Curvature bounds: discrete versus continuous spacesANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS:...
Transcript of Curvature bounds: discrete versus continuous spacesANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS:...
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Curvature bounds: discrete versus continuousspaces
Anca Bonciocat, IMAR Bucharest
Based on a joint work with K. T. Sturm
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Approach
1 based on mass transportation, following
K. T. Sturm, On the geometry of metric measure spaces I,II, Acta Math. 2006
J. Lott, C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. 2009
2 rough curvature bounds will depend on a real parameterh > 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Approach
1 based on mass transportation, following
K. T. Sturm, On the geometry of metric measure spaces I,II, Acta Math. 2006
J. Lott, C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. 2009
2 rough curvature bounds will depend on a real parameterh > 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Approach
1 based on mass transportation, following
K. T. Sturm, On the geometry of metric measure spaces I,II, Acta Math. 2006
J. Lott, C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. 2009
2 rough curvature bounds will depend on a real parameterh > 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Approach
1 based on mass transportation, following
K. T. Sturm, On the geometry of metric measure spaces I,II, Acta Math. 2006
J. Lott, C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. 2009
2 rough curvature bounds will depend on a real parameterh > 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Point of view - Coarse geometry
Coarse geometry studies the "large scale" properties of spaces.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Point of view - Coarse geometry
Coarse geometry studies the "large scale" properties of spaces.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Point of view - Coarse geometry
Coarse geometry studies the "large scale" properties of spaces.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Point of view - Coarse geometry
Coarse geometry studies the "large scale" properties of spaces.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.
We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.
(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′
0 between the supportsM0 := supp[m] ⊂ M and M ′
0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.
The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The L2-Wasserstein space
The L2-Wasserstein distance between two measures µ and νon M is defined as
dW (µ, ν) = inf
{(∫M×M
d2(x , y)dq(x , y)
)1/2
: q coupling of µ, ν
},
with the convention inf ∅ = ∞.
P2(M, d) :={ν :∫
M d2(o, x)dν(x) <∞ for some o ∈ M}
.
(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).
P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The L2-Wasserstein space
The L2-Wasserstein distance between two measures µ and νon M is defined as
dW (µ, ν) = inf
{(∫M×M
d2(x , y)dq(x , y)
)1/2
: q coupling of µ, ν
},
with the convention inf ∅ = ∞.
P2(M, d) :={ν :∫
M d2(o, x)dν(x) <∞ for some o ∈ M}
.
(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).
P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The L2-Wasserstein space
The L2-Wasserstein distance between two measures µ and νon M is defined as
dW (µ, ν) = inf
{(∫M×M
d2(x , y)dq(x , y)
)1/2
: q coupling of µ, ν
},
with the convention inf ∅ = ∞.
P2(M, d) :={ν :∫
M d2(o, x)dν(x) <∞ for some o ∈ M}
.
(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).
P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The L2-Wasserstein space
The L2-Wasserstein distance between two measures µ and νon M is defined as
dW (µ, ν) = inf
{(∫M×M
d2(x , y)dq(x , y)
)1/2
: q coupling of µ, ν
},
with the convention inf ∅ = ∞.
P2(M, d) :={ν :∫
M d2(o, x)dν(x) <∞ for some o ∈ M}
.
(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).
P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The L2-Wasserstein space
The L2-Wasserstein distance between two measures µ and νon M is defined as
dW (µ, ν) = inf
{(∫M×M
d2(x , y)dq(x , y)
)1/2
: q coupling of µ, ν
},
with the convention inf ∅ = ∞.
P2(M, d) :={ν :∫
M d2(o, x)dν(x) <∞ for some o ∈ M}
.
(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).
P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The relative entropy
Ent(ν|m) :=
∫
M ρ log ρdm , for ν = ρ ·m
+∞ , otherwise
We denote by P∗2(M, d,m) the subspace of measuresν ∈ P2(M, d,m) of finite entropy Ent(ν|m) <∞.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The relative entropy
Ent(ν|m) :=
∫
M ρ log ρdm , for ν = ρ ·m
+∞ , otherwise
We denote by P∗2(M, d,m) the subspace of measuresν ∈ P2(M, d,m) of finite entropy Ent(ν|m) <∞.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Riemannian case
Theorem (v.Renesse-Sturm 2005)
For any smooth connected Riemannian manifold M withintrinsic metric d and volume measure m and any K ∈ R thefollowing properties are equivalent :
1 Ricx(v , v) ≥ K |v |2 for x ∈ M and v ∈ Tx(M).2 The entropy Ent(·|m) is displacement K -convex on P2(M)
in the sense that for each geodesic γ : [0,1] → P2(M) andfor each t ∈ [0,1]
Ent(γ(t)|m) ≤ (1− t)Ent(γ(0)|m) + tEnt(γ(1)|m)
−K2
t(1− t) d2W (γ(0), γ(1)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Riemannian case
Theorem (v.Renesse-Sturm 2005)
For any smooth connected Riemannian manifold M withintrinsic metric d and volume measure m and any K ∈ R thefollowing properties are equivalent :
1 Ricx(v , v) ≥ K |v |2 for x ∈ M and v ∈ Tx(M).2 The entropy Ent(·|m) is displacement K -convex on P2(M)
in the sense that for each geodesic γ : [0,1] → P2(M) andfor each t ∈ [0,1]
Ent(γ(t)|m) ≤ (1− t)Ent(γ(0)|m) + tEnt(γ(1)|m)
−K2
t(1− t) d2W (γ(0), γ(1)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Curvature bounds for metric measure spaces
Definition (Sturm, Acta Math. 2006)
A metric measure space (M, d,m) has curvature ≥ K for somenumber K ∈ R iff the relative entropy Ent(·|m) is weaklyK -convex on P∗2(M, d,m) in the sense that for each pairν0, ν1 ∈ P∗2(M, d,m) there exists a geodesicΓ : [0,1] → P∗2(M, d,m) connecting ν0 and ν1 with
Ent(Γ(t)|m) ≤ (1− t)Ent(Γ(0)|m) + tEnt(Γ(1)|m)
−K2
t(1− t) d2W (Γ(0), Γ(1))
for all t ∈ [0,1].
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Curvature bounds for metric measure spaces
Definition (Sturm, Acta Math. 2006)
A metric measure space (M, d,m) has curvature ≥ K for somenumber K ∈ R iff the relative entropy Ent(·|m) is weaklyK -convex on P∗2(M, d,m) in the sense that for each pairν0, ν1 ∈ P∗2(M, d,m) there exists a geodesicΓ : [0,1] → P∗2(M, d,m) connecting ν0 and ν1 with
Ent(Γ(t)|m) ≤ (1− t)Ent(Γ(0)|m) + tEnt(Γ(1)|m)
−K2
t(1− t) d2W (Γ(0), Γ(1))
for all t ∈ [0,1].
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
L2-transportation distance D
D((M, d,m), (M ′, d′,m′)) = inf(∫
MtM′d̂
2(x , y)dq(x , y)
)1/2
,
where d̂ ranges over all couplings of d and d′ and q rangesover all couplings of m and m′.
A pseudo-metric d̂ on the disjoint union M tM ′ is a coupling ofd and d′ if d̂(x , y) = d(x , y) and d̂(x ′, y ′) = d′(x ′, y ′) for allx , y ∈ supp[m] ⊂ M and all x ′, y ′ ∈ supp[m′] ⊂ M ′.
D defines a complete separable length metric on the family ofall isomorphism classes of normalized metric measure spaces(M, d,m) with m ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
L2-transportation distance D
D((M, d,m), (M ′, d′,m′)) = inf(∫
MtM′d̂
2(x , y)dq(x , y)
)1/2
,
where d̂ ranges over all couplings of d and d′ and q rangesover all couplings of m and m′.
A pseudo-metric d̂ on the disjoint union M tM ′ is a coupling ofd and d′ if d̂(x , y) = d(x , y) and d̂(x ′, y ′) = d′(x ′, y ′) for allx , y ∈ supp[m] ⊂ M and all x ′, y ′ ∈ supp[m′] ⊂ M ′.
D defines a complete separable length metric on the family ofall isomorphism classes of normalized metric measure spaces(M, d,m) with m ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
L2-transportation distance D
D((M, d,m), (M ′, d′,m′)) = inf(∫
MtM′d̂
2(x , y)dq(x , y)
)1/2
,
where d̂ ranges over all couplings of d and d′ and q rangesover all couplings of m and m′.
A pseudo-metric d̂ on the disjoint union M tM ′ is a coupling ofd and d′ if d̂(x , y) = d(x , y) and d̂(x ′, y ′) = d′(x ′, y ′) for allx , y ∈ supp[m] ⊂ M and all x ′, y ′ ∈ supp[m′] ⊂ M ′.
D defines a complete separable length metric on the family ofall isomorphism classes of normalized metric measure spaces(M, d,m) with m ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
L2-transportation distance D
D((M, d,m), (M ′, d′,m′)) = inf(∫
MtM′d̂
2(x , y)dq(x , y)
)1/2
,
where d̂ ranges over all couplings of d and d′ and q rangesover all couplings of m and m′.
A pseudo-metric d̂ on the disjoint union M tM ′ is a coupling ofd and d′ if d̂(x , y) = d(x , y) and d̂(x ′, y ′) = d′(x ′, y ′) for allx , y ∈ supp[m] ⊂ M and all x ′, y ′ ∈ supp[m′] ⊂ M ′.
D defines a complete separable length metric on the family ofall isomorphism classes of normalized metric measure spaces(M, d,m) with m ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Let h > 0 be given. We say that a metric space (M, d) ish-rough geodesic iff for each pair of points x0, x1 ∈ M and eacht ∈ [0,1] there exists a point xt ∈ M satisfying
d(x0, xt) ≤ t d(x0, x1) + h
d(xt , x1) ≤ (1− t) d(x0, x1) + h
The point xt will be referred to as the h-rough t-intermediatepoint between x0 and x1.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Let h > 0 be given. We say that a metric space (M, d) ish-rough geodesic iff for each pair of points x0, x1 ∈ M and eacht ∈ [0,1] there exists a point xt ∈ M satisfying
d(x0, xt) ≤ t d(x0, x1) + h
d(xt , x1) ≤ (1− t) d(x0, x1) + h
The point xt will be referred to as the h-rough t-intermediatepoint between x0 and x1.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Let h > 0 be given. We say that a metric space (M, d) ish-rough geodesic iff for each pair of points x0, x1 ∈ M and eacht ∈ [0,1] there exists a point xt ∈ M satisfying
d(x0, xt) ≤ t d(x0, x1) + h
d(xt , x1) ≤ (1− t) d(x0, x1) + h
The point xt will be referred to as the h-rough t-intermediatepoint between x0 and x1.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.
2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).
3 For ε > 0 the space (Rn, d) with the metricd(x , y) =
√ε|x − y |+ |x − y |2 is h-rough geodesic for
each h ≥ ε/4.4 Discrete spaces, graphs.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.
2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).
3 For ε > 0 the space (Rn, d) with the metricd(x , y) =
√ε|x − y |+ |x − y |2 is h-rough geodesic for
each h ≥ ε/4.4 Discrete spaces, graphs.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.
2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).
3 For ε > 0 the space (Rn, d) with the metricd(x , y) =
√ε|x − y |+ |x − y |2 is h-rough geodesic for
each h ≥ ε/4.4 Discrete spaces, graphs.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.
2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).
3 For ε > 0 the space (Rn, d) with the metricd(x , y) =
√ε|x − y |+ |x − y |2 is h-rough geodesic for
each h ≥ ε/4.4 Discrete spaces, graphs.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.
2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).
3 For ε > 0 the space (Rn, d) with the metricd(x , y) =
√ε|x − y |+ |x − y |2 is h-rough geodesic for
each h ≥ ε/4.4 Discrete spaces, graphs.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Let (M, d) be a metric space. For each h > 0 and any pair ofmeasures ν0, ν1 ∈ P2(M, d) put
d±hW (ν0, ν1) := inf
{(∫[( d(x0, x1)∓h)+]2 dq(x0, x1)
)1/2},
where q ranges over all couplings of ν0 and ν1 and (·)+ denotesthe positive part.
The infimum above is attained. A coupling q for which theinfimum is attained in the definition of d±h
W is called ±h-optimalcoupling.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Let (M, d) be a metric space. For each h > 0 and any pair ofmeasures ν0, ν1 ∈ P2(M, d) put
d±hW (ν0, ν1) := inf
{(∫[( d(x0, x1)∓h)+]2 dq(x0, x1)
)1/2},
where q ranges over all couplings of ν0 and ν1 and (·)+ denotesthe positive part.
The infimum above is attained. A coupling q for which theinfimum is attained in the definition of d±h
W is called ±h-optimalcoupling.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
LemmaFor any 0 < h and 0 < h1 < h2 we have :
1 d+hW ≤ dW ≤ d+h
W + h ;2 dW ≤ d−h
W ≤ dW + h
3 d−h1W < d−h2
W
4 d+h1W (ν0, ν1) ≥ d+h2
W (ν0, ν1) with strict inequality if and onlyif d+h1
W (ν0, ν1) > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Rough curvature bounds
Definition
(M, d,m) has h-rough curvature ≥ K iff for each pairν0, ν1 ∈ P∗2(M, d,m) and for any t ∈ [0,1] there exists anh-rough t-intermediate point ηt ∈ P∗2(M, d,m) between ν0 andν1 satisfying
Ent(ηt |m) ≤ (1−t)Ent(ν0|m)+tEnt(ν1|m)−K2
t(1−t) d±hW (ν0, ν1)
2,
where the sign in d±hW (ν0, ν1) is chosen ’+’ if K > 0 and ’−’ if
K < 0.
Briefly, we write in this case h- Curv(M, d,m) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Rough curvature bounds
Definition
(M, d,m) has h-rough curvature ≥ K iff for each pairν0, ν1 ∈ P∗2(M, d,m) and for any t ∈ [0,1] there exists anh-rough t-intermediate point ηt ∈ P∗2(M, d,m) between ν0 andν1 satisfying
Ent(ηt |m) ≤ (1−t)Ent(ν0|m)+tEnt(ν1|m)−K2
t(1−t) d±hW (ν0, ν1)
2,
where the sign in d±hW (ν0, ν1) is chosen ’+’ if K > 0 and ’−’ if
K < 0.
Briefly, we write in this case h- Curv(M, d,m) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Rough curvature bounds
Definition
(M, d,m) has h-rough curvature ≥ K iff for each pairν0, ν1 ∈ P∗2(M, d,m) and for any t ∈ [0,1] there exists anh-rough t-intermediate point ηt ∈ P∗2(M, d,m) between ν0 andν1 satisfying
Ent(ηt |m) ≤ (1−t)Ent(ν0|m)+tEnt(ν1|m)−K2
t(1−t) d±hW (ν0, ν1)
2,
where the sign in d±hW (ν0, ν1) is chosen ’+’ if K > 0 and ’−’ if
K < 0.
Briefly, we write in this case h- Curv(M, d,m) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Properties
1 If (M, d,m) and (M ′, d′,m′) are isomorphic then
h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K
2 For 0 < h1 < h2
h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .
3 If (M, d,m) is a metric measure space and α, β > 0 then
h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Properties
1 If (M, d,m) and (M ′, d′,m′) are isomorphic then
h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K
2 For 0 < h1 < h2
h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .
3 If (M, d,m) is a metric measure space and α, β > 0 then
h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Properties
1 If (M, d,m) and (M ′, d′,m′) are isomorphic then
h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K
2 For 0 < h1 < h2
h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .
3 If (M, d,m) is a metric measure space and α, β > 0 then
h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Properties
1 If (M, d,m) and (M ′, d′,m′) are isomorphic then
h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K
2 For 0 < h1 < h2
h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .
3 If (M, d,m) is a metric measure space and α, β > 0 then
h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Properties
1 If (M, d,m) and (M ′, d′,m′) are isomorphic then
h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K
2 For 0 < h1 < h2
h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .
3 If (M, d,m) is a metric measure space and α, β > 0 then
h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Stability under convergence
TheoremLet (M, d,m) be a compact normalized metric measure spaceand consider {(Mh, dh,mh)}h>0 a family of normalized metricmeasure spaces with uniformly bounded diameter and withh- Curv(Mh, dh,mh) ≥ Kh for Kh → K as h → 0. If
(Mh, dh,mh)D−→ (M, d,m)
as h → 0 thenCurv(M, d,m) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Stability under convergence
TheoremLet (M, d,m) be a compact normalized metric measure spaceand consider {(Mh, dh,mh)}h>0 a family of normalized metricmeasure spaces with uniformly bounded diameter and withh- Curv(Mh, dh,mh) ≥ Kh for Kh → K as h → 0. If
(Mh, dh,mh)D−→ (M, d,m)
as h → 0 thenCurv(M, d,m) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Let (M, d,m) be a given metric measure space.
For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃
i=1BR(xi), where
R = R(h) ↘ 0 as h ↘ 0.
Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .
and∞⋃
i=1Ai = M.
Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.
We call (Mh, d,mh) a discretization of (M, d,m).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Discretizations of metric measure spaces
Theorem
1 If m(M) <∞ then (Mh, d,mh)D−→ (M, d,m) as h → 0.
2 If Curv(M, d,m) ≥ K for some real number K then foreach h > 0 and for each discretization (Mh, d,mh) withR(h) ≤ h/4 we have h- Curv(Mh, d,mh) ≥ K .
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Examples
1 Zn with the metric d1 coming from the norm|x |1 =
∑ni=1 |xi | and with the measure mn =
∑x∈Zn δx has
h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph
distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).
3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r
√3/3.
4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Examples
1 Zn with the metric d1 coming from the norm|x |1 =
∑ni=1 |xi | and with the measure mn =
∑x∈Zn δx has
h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph
distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).
3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r
√3/3.
4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Zn with the metric d1 coming from the norm|x |1 =
∑ni=1 |xi | and with the measure mn =
∑x∈Zn δx has
h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph
distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).
3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r
√3/3.
4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Zn with the metric d1 coming from the norm|x |1 =
∑ni=1 |xi | and with the measure mn =
∑x∈Zn δx has
h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph
distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).
3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r
√3/3.
4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Examples
1 Zn with the metric d1 coming from the norm|x |1 =
∑ni=1 |xi | and with the measure mn =
∑x∈Zn δx has
h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph
distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).
3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r
√3/3.
4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
G(l ,n, r) with vertices of constant degree l ≥ 3, with facesbounded by polygons with n ≥ 3 edges and with all edges ofthe same length r > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
G(l ,n, r) with vertices of constant degree l ≥ 3, with facesbounded by polygons with n ≥ 3 edges and with all edges ofthe same length r > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
G(l ,n, r) with vertices of constant degree l ≥ 3, with facesbounded by polygons with n ≥ 3 edges and with all edges ofthe same length r > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
1 If 1l + 1
n <12 then G(l ,n, r) can be embedded into the
2-dimensional hyperbolic space with constant sectionalcurvature
K = − 1r2
[arccosh
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
2 If 1l + 1
n >12 then G(l ,n, r) can be embedded into the
2-dimensional sphere with constant sectional curvature
K =1r2
[arccos
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
3 If 1l + 1
n = 12 then G(l ,n, r) can be embedded into the
euclidian plane (K = 0).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Homogeneous planar graphs
1 If 1l + 1
n <12 then G(l ,n, r) can be embedded into the
2-dimensional hyperbolic space with constant sectionalcurvature
K = − 1r2
[arccosh
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
2 If 1l + 1
n >12 then G(l ,n, r) can be embedded into the
2-dimensional sphere with constant sectional curvature
K =1r2
[arccos
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
3 If 1l + 1
n = 12 then G(l ,n, r) can be embedded into the
euclidian plane (K = 0).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Homogeneous planar graphs
1 If 1l + 1
n <12 then G(l ,n, r) can be embedded into the
2-dimensional hyperbolic space with constant sectionalcurvature
K = − 1r2
[arccosh
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
2 If 1l + 1
n >12 then G(l ,n, r) can be embedded into the
2-dimensional sphere with constant sectional curvature
K =1r2
[arccos
(2
cos2 (πn
)sin2 (π
l
) − 1
)]2
.
3 If 1l + 1
n = 12 then G(l ,n, r) can be embedded into the
euclidian plane (K = 0).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Homogeneous planar graphs
We equip G(l ,n, r) with the metric d induced by thecorresponding Riemannian metric and with the uniformmeasure m on the edges.
Proposition
(G(l ,n, r), d,m) has h-curvature ≥ K for h ≥ r · C(l ,n), where
K =
− 1r2
[arccosh
(2
cos2(πn )
sin2(πl )− 1)]2
for 1l + 1
n >12
1r2
[arccos
(2
cos2(πn )
sin2(πl )− 1)]2
for 1l + 1
n <12
0 for 1l + 1
n = 12
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
We equip G(l ,n, r) with the metric d induced by thecorresponding Riemannian metric and with the uniformmeasure m on the edges.
Proposition
(G(l ,n, r), d,m) has h-curvature ≥ K for h ≥ r · C(l ,n), where
K =
− 1r2
[arccosh
(2
cos2(πn )
sin2(πl )− 1)]2
for 1l + 1
n >12
1r2
[arccos
(2
cos2(πn )
sin2(πl )− 1)]2
for 1l + 1
n <12
0 for 1l + 1
n = 12
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Homogeneous planar graphs
Here C(l ,n) = 4 ·arcsinh
1
sin(πn )
scos2(π
n )
sin2(πl )−1
!
arccosh
2cos2(π
n )
sin2(πl )−1
!
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Transportation cost inequality
The probability measure m satisfies a Talagrand inequality (ortransportation cost inequality) with constant K iff for allν ∈ P2(M, d)
dW (ν,m) ≤√
2 Ent(ν|m)
K.
Proposition ("h-Talagrand Inequality")
Assume that (M, d,m) is a metric measure space which hash- Curv(M, d,m) ≥ K for some numbers h > 0 and K > 0.Then for each ν ∈ P2(M, d) we have
d+hW (ν,m) ≤
√2 Ent(ν|m)
K. (1.1)
We will call (1.1) h-Talagrand inequality.ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Transportation cost inequality
The probability measure m satisfies a Talagrand inequality (ortransportation cost inequality) with constant K iff for allν ∈ P2(M, d)
dW (ν,m) ≤√
2 Ent(ν|m)
K.
Proposition ("h-Talagrand Inequality")
Assume that (M, d,m) is a metric measure space which hash- Curv(M, d,m) ≥ K for some numbers h > 0 and K > 0.Then for each ν ∈ P2(M, d) we have
d+hW (ν,m) ≤
√2 Ent(ν|m)
K. (1.1)
We will call (1.1) h-Talagrand inequality.ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Transportation cost inequality
The probability measure m satisfies a Talagrand inequality (ortransportation cost inequality) with constant K iff for allν ∈ P2(M, d)
dW (ν,m) ≤√
2 Ent(ν|m)
K.
Proposition ("h-Talagrand Inequality")
Assume that (M, d,m) is a metric measure space which hash- Curv(M, d,m) ≥ K for some numbers h > 0 and K > 0.Then for each ν ∈ P2(M, d) we have
d+hW (ν,m) ≤
√2 Ent(ν|m)
K. (1.1)
We will call (1.1) h-Talagrand inequality.ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Transportation cost inequality
The probability measure m satisfies a Talagrand inequality (ortransportation cost inequality) with constant K iff for allν ∈ P2(M, d)
dW (ν,m) ≤√
2 Ent(ν|m)
K.
Proposition ("h-Talagrand Inequality")
Assume that (M, d,m) is a metric measure space which hash- Curv(M, d,m) ≥ K for some numbers h > 0 and K > 0.Then for each ν ∈ P2(M, d) we have
d+hW (ν,m) ≤
√2 Ent(ν|m)
K. (1.1)
We will call (1.1) h-Talagrand inequality.ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Concentration of measure
For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.
The concentration function of (M, d,m) is defined as
α(M,d,m)(r) := sup{
1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12
}, r > 0.
Proposition
Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0
α(M, d,m)(r) ≤ e−Kr2/8.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Concentration of measure
For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.
The concentration function of (M, d,m) is defined as
α(M,d,m)(r) := sup{
1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12
}, r > 0.
Proposition
Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0
α(M, d,m)(r) ≤ e−Kr2/8.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Concentration of measure
For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.
The concentration function of (M, d,m) is defined as
α(M,d,m)(r) := sup{
1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12
}, r > 0.
Proposition
Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0
α(M, d,m)(r) ≤ e−Kr2/8.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Concentration of measure
For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.
The concentration function of (M, d,m) is defined as
α(M,d,m)(r) := sup{
1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12
}, r > 0.
Proposition
Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0
α(M, d,m)(r) ≤ e−Kr2/8.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Concentration of measure
For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.
The concentration function of (M, d,m) is defined as
α(M,d,m)(r) := sup{
1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12
}, r > 0.
Proposition
Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0
α(M, d,m)(r) ≤ e−Kr2/8.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The Rényi entropy functional
SN(·|m) : P2(M, d) → R
withSN(ν|m) := −
∫Mρ−1/Ndν,
where ρ is the density of the absolutely continuous part νc withrespect to m in the Lebesgue decompositionν = νc + νs = ρm + νs of the measure ν ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The Rényi entropy functional
SN(·|m) : P2(M, d) → R
withSN(ν|m) := −
∫Mρ−1/Ndν,
where ρ is the density of the absolutely continuous part νc withrespect to m in the Lebesgue decompositionν = νc + νs = ρm + νs of the measure ν ∈ P2(M, d).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The Rényi entropy functional
Assume that m(M) is finite.
1 For each N > 1 the Rényi entropy functional SN(·|m) islower semicontinuous and satisfies
−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).
2 For any ν ∈ P2(M, d)
Ent(·|m) = limN→∞
N(1 + SN(ν|m)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The Rényi entropy functional
Assume that m(M) is finite.
1 For each N > 1 the Rényi entropy functional SN(·|m) islower semicontinuous and satisfies
−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).
2 For any ν ∈ P2(M, d)
Ent(·|m) = limN→∞
N(1 + SN(ν|m)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The Rényi entropy functional
Assume that m(M) is finite.
1 For each N > 1 the Rényi entropy functional SN(·|m) islower semicontinuous and satisfies
−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).
2 For any ν ∈ P2(M, d)
Ent(·|m) = limN→∞
N(1 + SN(ν|m)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The Rényi entropy functional
Assume that m(M) is finite.
1 For each N > 1 the Rényi entropy functional SN(·|m) islower semicontinuous and satisfies
−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).
2 For any ν ∈ P2(M, d)
Ent(·|m) = limN→∞
N(1 + SN(ν|m)).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
For given K ,N ∈ R with N ≥ 1 and (t , θ) ∈ [0,1]× R+ put
τ(t)K ,N(θ) =
∞ , if K θ2 ≥ (N − 1)π2
t1N
(sin“q
KN−1 tθ
”sin“q
KN−1 θ
”)1− 1
N
, if 0 < K θ2 < (N − 1)π2
t , if K θ2 = 0 orif K θ2 < 0 and N = 1
t1N
(sinh
“q−KN−1 tθ
”sinh
“q−KN−1 θ
”)1− 1
N
, if K θ2 < 0 and N > 1
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Curvature-dimension condition for metric spaces
Definition (Sturm, Acta Math. 2006)
(M, d,m) satisfies the curvature-dimension condition CD(K ,N)iff for each pair ν0, ν1 ∈ P2(M, d,m) there exist an optimalcoupling q of ν0, ν1 and a geodesic Γ : [0,1] → P2(M, d,m)connecting ν0, ν1 and with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ ( d(x0, x1)) · ρ
−1/N′
0 (x0)
+ τ(t)K ,N′( d(x0, x1)) · ρ
−1/N′
1 (x1)]
dq(x0, x1)
for all t ∈ [0,1] and all N ′ ≥ N. Here ρi denotes the densityfunctions of the absolutely continuous parts of νi with respect tom, i = 1,2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Curvature-dimension condition for metric spaces
Definition (Sturm, Acta Math. 2006)
(M, d,m) satisfies the curvature-dimension condition CD(K ,N)iff for each pair ν0, ν1 ∈ P2(M, d,m) there exist an optimalcoupling q of ν0, ν1 and a geodesic Γ : [0,1] → P2(M, d,m)connecting ν0, ν1 and with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ ( d(x0, x1)) · ρ
−1/N′
0 (x0)
+ τ(t)K ,N′( d(x0, x1)) · ρ
−1/N′
1 (x1)]
dq(x0, x1)
for all t ∈ [0,1] and all N ′ ≥ N. Here ρi denotes the densityfunctions of the absolutely continuous parts of νi with respect tom, i = 1,2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition
Let (M, d) be a metric space and h ≥ 0, t ∈ [0,1] given realnumbers.
1 xt is an h-rough t-intermediate point of x0 and x1 in M if{d(x0, xt) ≤ t d(x0, x1) + hd(xt , x1) ≤ (1− t) d(x0, x1) + h
2 xt is an h-rough t-intermediate point of x0 and x1 in thestrong sense if
(1− t) d(x0, xt)2 + t d(xt , x1)
2 ≤ t(1− t) d(x0, x1)2 + h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition
Let (M, d) be a metric space and h ≥ 0, t ∈ [0,1] given realnumbers.
1 xt is an h-rough t-intermediate point of x0 and x1 in M if{d(x0, xt) ≤ t d(x0, x1) + hd(xt , x1) ≤ (1− t) d(x0, x1) + h
2 xt is an h-rough t-intermediate point of x0 and x1 in thestrong sense if
(1− t) d(x0, xt)2 + t d(xt , x1)
2 ≤ t(1− t) d(x0, x1)2 + h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition
Let (M, d) be a metric space and h ≥ 0, t ∈ [0,1] given realnumbers.
1 xt is an h-rough t-intermediate point of x0 and x1 in M if{d(x0, xt) ≤ t d(x0, x1) + hd(xt , x1) ≤ (1− t) d(x0, x1) + h
2 xt is an h-rough t-intermediate point of x0 and x1 in thestrong sense if
(1− t) d(x0, xt)2 + t d(xt , x1)
2 ≤ t(1− t) d(x0, x1)2 + h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition
Let (M, d) be a metric space and h ≥ 0, t ∈ [0,1] given realnumbers.
1 xt is an h-rough t-intermediate point of x0 and x1 in M if{d(x0, xt) ≤ t d(x0, x1) + hd(xt , x1) ≤ (1− t) d(x0, x1) + h
2 xt is an h-rough t-intermediate point of x0 and x1 in thestrong sense if
(1− t) d(x0, xt)2 + t d(xt , x1)
2 ≤ t(1− t) d(x0, x1)2 + h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition
Definition
(M, d,m) satisfies h-CD(K ,N) (resp. h-CDs(K ,N)) iff for eachpair ν0, ν1 ∈ P2(M, d,m) there exists a δh-optimal coupling q ofν0, ν1 such that for any t ∈ [0,1] there exists an h-rought-intermediate point (resp. in the strong sense)ηt ∈ P2(M, d,m) of ν0, ν1 with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′
0 (x0)
+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′
1 (x1)]
dq(x0, x1)
for all N ′ ≥ N.
ρi = the density of the absolutely continuous part of νi w.r.t. m.δ = −1 for K < 0 and δ = 1 for K ≥ 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition
Definition
(M, d,m) satisfies h-CD(K ,N) (resp. h-CDs(K ,N)) iff for eachpair ν0, ν1 ∈ P2(M, d,m) there exists a δh-optimal coupling q ofν0, ν1 such that for any t ∈ [0,1] there exists an h-rought-intermediate point (resp. in the strong sense)ηt ∈ P2(M, d,m) of ν0, ν1 with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′
0 (x0)
+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′
1 (x1)]
dq(x0, x1)
for all N ′ ≥ N.
ρi = the density of the absolutely continuous part of νi w.r.t. m.δ = −1 for K < 0 and δ = 1 for K ≥ 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition
Definition
(M, d,m) satisfies h-CD(K ,N) (resp. h-CDs(K ,N)) iff for eachpair ν0, ν1 ∈ P2(M, d,m) there exists a δh-optimal coupling q ofν0, ν1 such that for any t ∈ [0,1] there exists an h-rought-intermediate point (resp. in the strong sense)ηt ∈ P2(M, d,m) of ν0, ν1 with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′
0 (x0)
+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′
1 (x1)]
dq(x0, x1)
for all N ′ ≥ N.
ρi = the density of the absolutely continuous part of νi w.r.t. m.δ = −1 for K < 0 and δ = 1 for K ≥ 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition
Definition
(M, d,m) satisfies h-CD(K ,N) (resp. h-CDs(K ,N)) iff for eachpair ν0, ν1 ∈ P2(M, d,m) there exists a δh-optimal coupling q ofν0, ν1 such that for any t ∈ [0,1] there exists an h-rought-intermediate point (resp. in the strong sense)ηt ∈ P2(M, d,m) of ν0, ν1 with
SN′(ηt |m) ≤ −∫ [
τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′
0 (x0)
+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′
1 (x1)]
dq(x0, x1)
for all N ′ ≥ N.
ρi = the density of the absolutely continuous part of νi w.r.t. m.δ = −1 for K < 0 and δ = 1 for K ≥ 0
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition
For K = 0 the above inequality reads
SN′(ηt |m) ≤ (1− t) · SN′(ν0|m) + t · SN′(ν1|m),
h-CD(0,N) requires the Rényi entropy functionals SN′(·|m) tobe weakly convex on P2(M, d,m) along "h-rough geodesics" forall N ′ ≥ N.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition
For K = 0 the above inequality reads
SN′(ηt |m) ≤ (1− t) · SN′(ν0|m) + t · SN′(ν1|m),
h-CD(0,N) requires the Rényi entropy functionals SN′(·|m) tobe weakly convex on P2(M, d,m) along "h-rough geodesics" forall N ′ ≥ N.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
The rough curvature-dimension condition - Properties
Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :
1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.
2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).
3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.
4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).
The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
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Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
(M, d,m) metric measure space
µ0, µ1 probability measures, Ai := supp[µi ]
η h-rough t-intermediate point in the strong sense of µ0 and µ1
For λ ≥ 0 denote
Aλt :=
{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2
+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)
2 + λ2)}.
Then the following estimate holds :
η({Aλt ) ≤ h2/λ2 for any λ > 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
Moreover, if 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λi ≤ . . . then
∞∑i=1
λ2i · η(A
λi+1t \ Aλi
t ) ≤ h2
or, equivalently,
∞∑i=1
η({Aλit )(λ2
i − λ2i−1) ≤ h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences
Moreover, if 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λi ≤ . . . then
∞∑i=1
λ2i · η(A
λi+1t \ Aλi
t ) ≤ h2
or, equivalently,
∞∑i=1
η({Aλit )(λ2
i − λ2i−1) ≤ h2.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
The classical Brunn-Minkowski in Rn :
voln(tA + (1− t)B)1/n ≥ t voln(A)1/n + (1− t)voln(B)1/n
for all bounded Borel measurable subsets A and B and anyt ∈ [0,1].
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
The classical Brunn-Minkowski in Rn :
voln(tA + (1− t)B)1/n ≥ t voln(A)1/n + (1− t)voln(B)1/n
for all bounded Borel measurable subsets A and B and anyt ∈ [0,1].
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
Proposition
Let (M, d,m) be a metric measure space that has finite massand satisfies h-CDs(K ,N) for some numbers h ≥ 0, K ,N ∈ R,N ≥ 1. Then for any measurable sets A0, A1 ⊂ M withm(A0) ·m(A1) > 0, for any t ∈ [0,1] , N ′ ≥ N and any λ > 0
m(Aλt )1/N′
+(h2/λ2)1−1/N′m({Aλ
t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)
1/N′
+τ(t)K ,N′(Θh) · (A1)
1/N′,
with Aλt as before and Θh is given by
Θh :=
{infx0∈A0,a1∈A1( d(x0, x1)− h)+, if K ≥ 0supx0∈A0,a1∈A1
( d(x0, x1) + h), if K < 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
Proposition
Let (M, d,m) be a metric measure space that has finite massand satisfies h-CDs(K ,N) for some numbers h ≥ 0, K ,N ∈ R,N ≥ 1. Then for any measurable sets A0, A1 ⊂ M withm(A0) ·m(A1) > 0, for any t ∈ [0,1] , N ′ ≥ N and any λ > 0
m(Aλt )1/N′
+(h2/λ2)1−1/N′m({Aλ
t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)
1/N′
+τ(t)K ,N′(Θh) · (A1)
1/N′,
with Aλt as before and Θh is given by
Θh :=
{infx0∈A0,a1∈A1( d(x0, x1)− h)+, if K ≥ 0supx0∈A0,a1∈A1
( d(x0, x1) + h), if K < 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
Proposition
Let (M, d,m) be a metric measure space that has finite massand satisfies h-CDs(K ,N) for some numbers h ≥ 0, K ,N ∈ R,N ≥ 1. Then for any measurable sets A0, A1 ⊂ M withm(A0) ·m(A1) > 0, for any t ∈ [0,1] , N ′ ≥ N and any λ > 0
m(Aλt )1/N′
+(h2/λ2)1−1/N′m({Aλ
t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)
1/N′
+τ(t)K ,N′(Θh) · (A1)
1/N′,
with Aλt as before and Θh is given by
Θh :=
{infx0∈A0,a1∈A1( d(x0, x1)− h)+, if K ≥ 0supx0∈A0,a1∈A1
( d(x0, x1) + h), if K < 0.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
Corollary (’Generalized Brunn-Minkowski Inequality’)
Assume that (M, d,m) is a normalized metric measure spacethat satisfies h-CDs(K ,N). Then for any measurable sets A0,A1 ⊂ M with m(A0) ·m(A1) > 0, for any t ∈ [0,1] and N ′ ≥ N
m(A√
ht )1/N′
+h1−1/N′ ≥ τ(1−t)K ,N′ (Θh)m(A0)
1/N′+τ
(t)K ,N′(Θh)m(A1)
1/N′,
with Θh given above.In particular, if K ≥ 0 then
m(A√
ht )1/N′
+h1−1/N′ ≥ (1− t) ·m(A0)1/N′
+ t ·m(A1)1/N′
.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Brunn-Minkowski ineq.
Corollary (’Generalized Brunn-Minkowski Inequality’)
Assume that (M, d,m) is a normalized metric measure spacethat satisfies h-CDs(K ,N). Then for any measurable sets A0,A1 ⊂ M with m(A0) ·m(A1) > 0, for any t ∈ [0,1] and N ′ ≥ N
m(A√
ht )1/N′
+h1−1/N′ ≥ τ(1−t)K ,N′ (Θh)m(A0)
1/N′+τ
(t)K ,N′(Θh)m(A1)
1/N′,
with Θh given above.In particular, if K ≥ 0 then
m(A√
ht )1/N′
+h1−1/N′ ≥ (1− t) ·m(A0)1/N′
+ t ·m(A1)1/N′
.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Geometrical consequences - Bonnet-Myers Theorem
Corollary
For every normalized metric measure space (M, d,m) thatsatisfies the rough curvature-dimension condition h-CDs(K ,N)for some real numbers h > 0, K > 0 and N ≥ 1, the support ofthe measure m has diameter
L ≤√
N − 1K
π+h.
In particular, if K > 0 and N = 1 then supp[m] consists of a ballof radius h.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Stability under convergence
TheoremLet (M, d,m) be a normalized metric measure space and{(Mh, dh,mh)}h>0 a family of normalized metric measurespaces such that for each h > 0 the space (Mh, dh,mh)satisfies h-CD(Kh,Nh) and has diameter Lh for some realnumbers Kh,Nh and Lh with Nh ≥ 1 and Lh > 0. Assume that
(Mh, dh,mh)D−→ (M, d,m)
and(Kh,Nh,Lh) → (K ,N,L)
as h → 0 for some (K ,N,L) ∈ R3 satisfying K · L2 < (N − 1)π2.Then the space (M, d,m) satisfies the curvature-dimensioncondition CD(K ,N) and has diameter ≤ L.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Stability under convergence
TheoremLet (M, d,m) be a normalized metric measure space and{(Mh, dh,mh)}h>0 a family of normalized metric measurespaces such that for each h > 0 the space (Mh, dh,mh)satisfies h-CD(Kh,Nh) and has diameter Lh for some realnumbers Kh,Nh and Lh with Nh ≥ 1 and Lh > 0. Assume that
(Mh, dh,mh)D−→ (M, d,m)
and(Kh,Nh,Lh) → (K ,N,L)
as h → 0 for some (K ,N,L) ∈ R3 satisfying K · L2 < (N − 1)π2.Then the space (M, d,m) satisfies the curvature-dimensioncondition CD(K ,N) and has diameter ≤ L.
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Stability under discretization
TheoremLet (M, d,m) be a metric measure space that satisfies thecurvature-dimension condition CD(K ,N) for some realnumbers K and N ≥ 1. Then for each h > 0 any discretization(Mh, d,mh) with R(h) ≤ h/4 satisfies the roughcurvature-dimension condition h-CD(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Stability under discretization
TheoremLet (M, d,m) be a metric measure space that satisfies thecurvature-dimension condition CD(K ,N) for some realnumbers K and N ≥ 1. Then for each h > 0 any discretization(Mh, d,mh) with R(h) ≤ h/4 satisfies the roughcurvature-dimension condition h-CD(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
Stability under discretization
TheoremLet (M, d,m) be a metric measure space that satisfies thecurvature-dimension condition CD(K ,N) for some realnumbers K and N ≥ 1. Then for each h > 0 any discretization(Mh, d,mh) with R(h) ≤ h/4 satisfies the roughcurvature-dimension condition h-CD(K ,N).
ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES
ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN
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ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES