Transference Theorems in the Geometry of Numbers
description
Transcript of Transference Theorems in the Geometry of Numbers
Transference Theorems in the Geometry of Numbers
Daniel DadushNew York University
EPIT 2013
Convex body . (convex, full dimensional and bounded).
Convex Bodies
𝐾𝑥
𝑦
Non convex set.
Convexity: Line between and in .Equivalently
𝑥𝑦
𝐾 1
Input: Classic NP-Hard problem (integrality makes it hard)
IP Problem: Decide whether above system has a solution.
Focus for this talk: Geometry of Integer Programs
Integer Programming Problem (IP)
ℤ𝑛𝐾 2
convex set
𝐾 1
Input: (integrality makes it hard)
LP Problem: Decide whether above system has a solution.
Polynomial Time Solvable: Khachiyan `79 (Ellipsoid Algorithm)
Integer Programming Problem (IP)Linear Programming (LP)
ℤ𝑛𝐾 2
convex set
Input: : Invertible Transformation Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
𝐵𝐾1
𝐵ℤ𝑛
𝐵𝐾 2
Input: Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
𝐾 1
𝐿𝐾 2
𝐾 1
1) When can we guarantee that a convex set contains a lattice point? (guarantee IP feasibility)
2) What do lattice free convex sets look like? (sets not containing integer points)
Central Geometric Questions
ℤ2𝐾 2
𝐾 1
Examples
ℤ2
If a convex set very ``fat’’, then it will always contain a lattice point.
“Hidden cube”
𝐾 1
Examples
ℤ2
If a convex set very ``fat’’, then it will always contain a lattice point.
Examples
ℤ2
Volume does NOT guarantee lattice points (in contrast with Minkowski’s theorem).
Infinite band
Examples
ℤ2
However, lattice point free sets must be ``flat’’ in some direction.
Lattice Width
ℤ2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
𝑥1=1 𝑥1=8𝑥1=4 …
Lattice Width
ℤ2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
𝑥2=1𝑥2=2
𝑥2=3
𝑥2=4
Lattice Width
ℤ2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
Note: axis parallel hyperperplanes do NOT suffice.
𝑥1−𝑥2=1 𝑥1−𝑥2= 4
Lattice Width
ℤ2
Why is this useful? IP feasible regions: hyperplane decomposition enables reduction into dimensional sub-IPs.
# intersections # subproblems
𝑥2=1𝑥2=2
𝑥2=3
𝑥2=4
𝐾 2
𝐾 1
subproblems
subproblems
Lattice Width
ℤ2
Why is this useful? # intersections # subproblems
If # intersections is small, can solve IP via recursion.
𝑥2=1𝑥2=2
𝑥2=3
𝑥2=4
𝐾 2
𝐾 1
subproblems
subproblems
Lattice Width
Integer Hyperplane : Hyperplane where
Fact: is an integer hyperplane , ( called primitive if )
ℤ2
𝑥1−𝑥2=1 𝑥1−𝑥2= 4
𝐻 (1 ,−1)𝑎
Lattice Width
Hyperplane Decomposition of : For
(parallel hyperplanes)
ℤ2
𝑥1=1 𝑥1=8
(1 ,0)𝑎
…
Lattice Width
Hyperplane Decomposition of : For
(parallel hyperplanes)
Note: If is not primitive, decomposition is finer than necessary.
ℤ2
2 𝑥1=2 2 𝑥1=16
(2 ,0)2𝑎
2 𝑥1=5 2 𝑥1=9……
𝐾
Lattice Width
How many intersections with ? (parallel hyperplanes) # INTs }| + 1 (tight within +2)
ℤ2
𝑥1=1 𝑥1=8
(1 ,0)𝑎𝑥𝑚𝑎𝑥
𝑥𝑚𝑖𝑛
2 3 4 6 71.9
7.2
Lattice Width
Width Norm of : for any Lattice Width: width
ℤ2
𝐾 𝑦width𝐾 ( 𝑦 )=1.2
Kinchine’s Flatness Theorem
Theorem: For a convex body , , .
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Bounds improvements:Khinchine `48: Babai `86: Lenstra-Lagarias-Schnorr `87: Kannan-Lovasz `88: Banaszczyk et al `99: Rudelson `00:
𝐾
Properties of
Width Norm of : for any
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
𝐾 −𝐾
2𝑅𝐵2𝑛
Convex & Centrally Symmetric 2𝑟 𝐵2
𝑛
𝐾
Properties of
Width Norm of : for any Bounds:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Bounds:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Symmetry: By symmetry of
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Symmetry: Therefore
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Homogeneity: For (Trivial)
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
Properties of
is invariant under translations of .
𝐾
𝑦
𝑚𝑖𝑛 𝑚𝑎𝑥𝑤𝑖𝑑𝑡 h𝐾 (𝑦 )=𝑚𝑎𝑥−𝑚𝑖𝑛
Properties of
is invariant under translations of .
𝐾
𝑦
𝑚𝑖𝑛 𝑚𝑎𝑥
𝐾 + 𝑡
𝑦
𝑚𝑖𝑛+ ⟨𝑦 , 𝑡 ⟩ +
Properties of
is invariant under translations of .Also follows since .(width only looks at differences between vectors of .
𝐾
𝑦
𝑚𝑖𝑛 𝑚𝑎𝑥
𝐾 + 𝑡
𝑦
𝑚𝑖𝑛+ ⟨𝑦 , 𝑡 ⟩ +
Kinchine’s Flatness Theorem
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Remark: Finding flatness direction is a general norm SVP!
Theorem: For a convex body , such that , .
Kinchine’s Flatness Theorem
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Easy generalize to arbitrary lattices. (note )
Theorem: For a convex body , such that , .
Kinchine’s Flatness Theorem
Theorem: For a convex body , such that , .
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Easy generalize to arbitrary lattices. (note )
Kinchine’s Flatness Theorem
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Easy generalize to arbitrary lattices.
where is dual lattice.
Theorem: For a convex body and lattice , such that , .
Kinchine’s Flatness Theorem
Theorem: For a convex body and lattice , such that , .
[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]
Bound conjectured to be (best possible).
Homegeneity of Lattice Width:
Lower Bound: Simplex
Bound cannot be improved to .
(interior of S)Pf: If and , then a contradiction.
𝑛𝑒1
𝑛𝑒20
No interior lattice points.
𝑆
Lower Bound: Simplex
Bound cannot be improved to .
Pf: For , then
𝑛𝑒1
𝑛𝑒20𝑆 ℤ2
Flatness Theorem
Theorem*: For a convex body and lattice , if such that , then .
By shift invariance of .
𝐾𝐾 + 𝑡
ℤ2
Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
𝐾𝐾 + 𝑡
ℤ2
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾
Covering Radius
Definition: Covering radius of with respect to .
ℤ2
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 Condition from
Flatness Theorem
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾Condition fromFlatness Theorem𝑥
Covering Radius
Definition: Covering radius of with respect to .
ℤ22 (𝐾−𝑥 )+𝑥𝑥Must scale by factor about to hit .
Therefore .
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
Covering Radius
Definition: Further Properties:
ℤ2𝐾
Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
𝐾𝐾 + 𝑡
ℤ2
𝐾𝐾 + 𝑡
Flatness Theorem
Theorem: For a convex body and lattice :
By homogeneity of and .
ℤ2
Flatness Theorem
Theorem [Banaszczyk `93]: For a lattice in :
(since -=)
Bound Improvements:Khinchine `48: Babai `86: Lenstra-Lagarias-Schnorr `87: Kannan-Lovasz `88: Banaszczyk `93: (asymptotically optimal)
Flatness Theorem
Theorem [Banaszczyk `93]: For a lattice in :
𝐿𝑥𝜇
is max distance to
Flatness Theorem
Theorem [Banaszczyk `96]: For a symmetric convex body :
Bound Improvements:Khinchine `48: Babai `86: Kannan-Lovasz `88: Banaszczyk `93: Banaszczyk `96:
ℤ2𝐾
Lower Bounds: Random Lattices
Theorem: For a convex body and ``random’’ lattice in :
(Minkowski Hlawka + Volume Product Bound)
ℤ2𝐾
Symmetric convex body (.
Gauge function:
0
𝑥𝐾 𝑠𝐾
Norms and Convex Bodies
-
1. (triangle inequality) 2. (homogeneity)3. (symmetry)
is unit ball of
Convex body containing origin in its interior.
Gauge function: 𝑥𝐾 𝑠𝐾
Norms and Convex Bodies
0
1. (triangle inequality) 2. (homogeneity)3. (symmetry)
is unit ball of
Gauge function:
Triangle Inequality:
Want to show
By definition . May assume Need to show .
Norms and Convex Bodies
convex combination
asymmetric norm in . Unit ball .
𝑥
𝐾
Norms and Convex Bodies
0
is convex: Take
𝑦-
𝐾
Symmetric convex body and lattice in .If , such that .
Minkowski’s Convex Body Theorem
0
𝐾
Symmetric convex body and lattice in .If , such that .
Minkowski’s Convex Body Theorem
0 𝑦
𝐾
Pf: basis for with parallelepiped .
Minkowski’s Convex Body Theorem
0𝑏1
𝑏22𝑏2
2𝑏1𝑃
2𝑃
𝐾
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowski’s Convex Body Theorem
0
2𝑃
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
Pf: basis for . Let be parallelepiped. Shift tiles intersecting into using .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowski’s Convex Body Theorem
0𝐾
2𝑃𝑐
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
+
+ 𝑐
Here (a) , (b) , (c)
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
+
+ 𝑐
Then (by symmetry of ) and .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃𝑐
+
+
2(𝑦−𝑥)
Then (by symmetry of ) and .
Minkowski’s Convex Body Theorem
𝐾0
2(𝑦−𝑥)
2𝐾
So and
Minkowski’s Convex Body Theorem
𝐾0𝑦−𝑥
Symmetric convex body and lattice in . Successive Minima
0𝐾
𝜆1𝐾- 𝑦 1
Symmetric convex body and lattice in . Successive Minima
0
𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in . Successive Minima
𝑦 2𝜆2𝐾
-
0
𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in .Minkowski’s First Theorem
0
𝑠𝐾
Symmetric convex body and lattice in .Pf: Let . For
Minkowski’s First Theorem
0
𝑦
Symmetric convex body and lattice in .Pf: By Minkowski’s convex body theorem, .
Since this holds as , .
Minkowski’s First Theorem
0𝑠𝐾
𝑦 2𝜆2𝐾
-
𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in .Minkowski’s Second Theorem
0
Theorem [Kannan-Lovasz 88]:
For symmetric becomes
Covering Radius vs Successive Minima
“Naïve” Babai rounding
For a closed convex set and , there exists such that
Separator Theorem
Can use where is the closest point in to .
𝑝𝐾𝑦
𝑥∗
For a closed convex set and , there exists such that
𝑝𝐾
Separator Theorem
𝑦
If not separator can get closer to on line segment .
𝑧𝑥∗∗𝑥∗
For compact convex with in relative interior,the polar is
For , the dual norm
Remark:
Polar Bodies and Dual Norms
0(1,1)
(1,-1)
(-1,1)
(-1,-1) 0 (1,0)(-1,0)(0,1)
(0,-1)
Theorem: compact convex with in rel. int., then .In particular .
Polar Bodies and Dual NormsDef:
𝐾0
Theorem: compact convex with in rel. int., then .In particular .
Polar Bodies and Dual NormsDef:
Pf: Easy to check that.
Hence .
Must show .
𝐾0
Theorem: compact convex with in rel. int., .In particular .
Polar Bodies and Dual NormsDef:
Pf: Take and in .
By separator theorem such that
Scale such that
𝑝𝐾𝑦0
Then and therefore .
Theorem: compact convex with in rel. int., .In particular .
𝑝𝐾
Polar Bodies and Dual Norms
𝑦0
Def:
Theorem [Banaszczyk `95,`96]: For a symmetric convex body and lattice in
Pf of lower bnd: Take linearly independent vectors where , and where
Since , s.t. . Hence
Banaszczyk’s Transference Theorem
Theorem [Blashke `18, Santalo `49] : For a symmetric convex body
Theorem [Bourgain-Milman `87, Kuperberg `08]:For a symmetric convex body
Mahler Conjecture: minimized when is cube.
Volume Product Bounds
Theorem [Kannan-Lovasz `88] : For a symmetric convex body and lattice
Pf: By Minkowski’s first theorem
(Bourgain-Milman)
Minkowski Transference
a symmetric convex body and lattice L. Let the orthogonal projection onto a subspace .
Then for any
Also following identities hold:
and .
Projected Norms and Lattices
𝐾𝐾 + 𝑡
Flatness Theorem
We will follow proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :
ℤ2
Flatness Theorem
We will follow proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :
Proof of lower bound:
𝐾𝐾 + 𝑡
ℤ2
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: Let . Can write , .
Note that since .
By shifting , can assume that and (all quantities are invariant under shifts of )
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: By shifting , have and
𝐾ℤ2
𝑏1𝜆 𝑧 1
𝜆 𝑧 2
𝐾
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: By shifting , have and
ℤ2𝜆𝐾𝑏1
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: Let . Note that .Suffices to show that that
Let , By induction , hence such .
𝐾
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: By induction , hence such .
ℤ2
𝑏1
𝑏2𝑥𝑥
𝑦
𝜋 2
𝐾
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: Have such .By definition can find s.t. .
Since , we have , for some shift . Therefore , such that .
𝐾
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: By definition can find s.t. .
ℤ2
𝛼𝑏1𝑥𝑥
𝑦
𝜋 2
𝐾
�̂�
𝑏1
Generalized Babai Rounding
Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then
where (Gram-Schmidt Orth.) and Pf: Since , we have , for some shift .
Therefore , such that .
Since , note that Hence
Generalized HKZ Basis
For a symmetric convex body and lattice in
A generalized HKZ basis for with respect to satisfies where is orthogonal projection onto .
Flatness Theorem
Theorem: For a convex body and lattice :
Pf: Let be a HKZ basis with respect to . Pick j such that .
By generalized Babai,
By definition , hence
Flatness Theorem
Theorem: For a convex body and lattice :
For , we have and .
By the Minkowski transference
Flatness Theorem
Theorem: For a convex body and lattice :
By the Minkowski transference
By inclusion .Hence
𝐾
𝐾𝐾
Theorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
Subspace Flatness Theorem
ℤ2
𝐾𝐾
Theorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
Subspace Flatness Theorem
ℤ2
𝐾
Theorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
Subspace Flatness Theorem
ℤ2
𝐾
: r=2
0
𝐻={0 }
ℤ2
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
: r=2
0
Integral shifts of
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
: r=2
Integral shifts of intersecting
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
𝐻={( x , y ) : y=0 }
0
: r=1
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
Integral shifts of
0
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
Integral shifts of intersecting
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
If , is a hyperplane.Forcing corresponds to Classical Flatness Theorem.
ℤ2
Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
𝐾
ℤ2
Subspace Flatness TheoremConjecture [KL `88, D.12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .
Inhomogeneous Minkowski
Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Pf: Let be a HKZ basis with respect to with satisfying as before.
Inhomogeneous Minkowski
Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Pf: Let and . By Minkowski’s first theorem
.
Gaussian Heuristic
Lemma: For a convex body and lattice , and , then for any
Pf: By shift invariance may assume . Since , can choose fundamental domain .Here tiles space wrt to and.
.
Hence
Subspace Flatness
Corollary: Convex body and lattice . Assume for . a subspace , of some dimension , such that
Pf: May assume , since this is worst case.Picking from inhomogeneous Minkowski theorem, we have
By Gaussian Heuristic, for any , .
Subspace Flatness
Conjecture [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Corollary of Conjecture: Convex body and lattice . Assume for . a subspace , of some dimension , such that
Generalized HKZ Basis
For a symmetric convex body and lattice in
A generalized HKZ basis for with respect to satisfies where is orthogonal projection onto .
Inhomogeneous Minkowski
Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Hence having ``large’’ volume (i.e. relative to determinant) in every projection implies ``always’’ contains lattice points.
In this sense, we get a generalization of Minkowski’s theorem for arbitrary convex bodies.
Inhomogeneous Minkowski
Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Pf: Let be a HKZ basis with respect to with satisfying as before.
Inhomogeneous Minkowski
Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Pf: Let and . By Generalized Babai and Minkowski’s first theorem
.
Brunn-Minkowski: +
Gaussian Heuristic
Lemma: For a convex body and lattice , and , then for any
𝐾 +𝑥
Gaussian HeuristicWant to bound .By shifting may assume .
𝐿
𝐾
Gaussian HeuristicSince covers space, exists fundamental domain .
𝐹𝐿
𝐿
𝐾
Gaussian HeuristicPlace around each point in .
𝐹
𝐿
𝐾
Gaussian HeuristicPlace around each point in .
Hence
𝐹
Subspace Flatness Theorem
Corollary: Convex body and lattice . Assume for . a subspace , of some dimension , such that
Pf: May assume , since this is worst case.Picking from inhomogeneous Minkowski theorem, we have
By Gaussian Heuristic, for any , .
𝐿
12 𝐾
Subspace FlatnessHave .
𝐾
𝐿
Subspace FlatnessHence can find projection such that
is small.
𝐾
𝐿
𝐾
Subspace FlatnessCan find projection such that
is small.
𝑊
𝜋𝑊 (𝐾 )
𝜋𝑊 (𝐿)
𝐿
𝐾
Subspace FlatnessCorresponds to small number of shifts
of intersecting .
𝑊
𝜋𝑊 (𝐾 )
𝜋𝑊 (𝐿)
+ +…
Subspace Flatness Theorem
Conjecture [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that
Corollary of Conjecture: Convex body and lattice . Assume for . a subspace , of some dimension , such that
Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Lower bound valid for all . Given lower bound is polytime computable.
Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Promise problem: -coGapCRP (Covering Radius Problem) where
YES instances:
No instances:
Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Promise problem: -coGapCRP (Covering Radius Problem)
Conjecture implies -coGapCRP NP.
Current best: -coGapCRP NP. (Exponential Improvement!)