Transference Theorems in the Geometry of Numbers
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Transference Theorems in the Geometry of Numbers
Daniel DadushNew York University
EPIT 2013
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Convex body . (convex, full dimensional and bounded).
Convex Bodies
𝐾𝑥
𝑦
Non convex set.
Convexity: Line between and in .Equivalently
𝑥𝑦
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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
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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
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Input: : Invertible Transformation Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
𝐵𝐾1
𝐵ℤ𝑛
𝐵𝐾 2
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Input: Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
𝐾 1
𝐿𝐾 2
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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
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𝐾 1
Examples
ℤ2
If a convex set very ``fat’’, then it will always contain a lattice point.
“Hidden cube”
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𝐾 1
Examples
ℤ2
If a convex set very ``fat’’, then it will always contain a lattice point.
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Examples
ℤ2
Volume does NOT guarantee lattice points (in contrast with Minkowski’s theorem).
Infinite band
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Examples
ℤ2
However, lattice point free sets must be ``flat’’ in some direction.
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Lattice Width
ℤ2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
𝑥1=1 𝑥1=8𝑥1=4 …
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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
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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
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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
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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
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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
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Lattice Width
Hyperplane Decomposition of : For
(parallel hyperplanes)
ℤ2
𝑥1=1 𝑥1=8
(1 ,0)𝑎
…
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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……
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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
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Lattice Width
Width Norm of : for any Lattice Width: width
ℤ2
𝐾 𝑦width𝐾 ( 𝑦 )=1.2
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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:
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𝐾
Properties of
Width Norm of : for any
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
𝐾 −𝐾
2𝑅𝐵2𝑛
Convex & Centrally Symmetric 2𝑟 𝐵2
𝑛
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𝐾
Properties of
Width Norm of : for any Bounds:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Bounds:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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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
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𝐾
Properties of
Width Norm of : for any Symmetry: Therefore
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Homogeneity: For (Trivial)
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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𝐾
Properties of
Width Norm of : for any Triangle Inequality:
𝑟 𝐵2𝑛
𝑡
𝑅𝐵2𝑛 0
2𝑟 𝐵2𝑛
2𝑅𝐵2𝑛
Convex & Centrally Symmetric
𝐾 −𝐾
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Properties of
is invariant under translations of .
𝐾
𝑦
𝑚𝑖𝑛 𝑚𝑎𝑥𝑤𝑖𝑑𝑡 h𝐾 (𝑦 )=𝑚𝑎𝑥−𝑚𝑖𝑛
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Properties of
is invariant under translations of .
𝐾
𝑦
𝑚𝑖𝑛 𝑚𝑎𝑥
𝐾 + 𝑡
𝑦
𝑚𝑖𝑛+ ⟨𝑦 , 𝑡 ⟩ +
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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 ⟩ +
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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 , .
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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 , .
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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 )
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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 , .
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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:
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Lower Bound: Simplex
Bound cannot be improved to .
(interior of S)Pf: If and , then a contradiction.
𝑛𝑒1
𝑛𝑒20
No interior lattice points.
𝑆
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Lower Bound: Simplex
Bound cannot be improved to .
Pf: For , then
𝑛𝑒1
𝑛𝑒20𝑆 ℤ2
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Flatness Theorem
Theorem*: For a convex body and lattice , if such that , then .
By shift invariance of .
𝐾𝐾 + 𝑡
ℤ2
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Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
𝐾𝐾 + 𝑡
ℤ2
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 Condition from
Flatness Theorem
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾Condition fromFlatness Theorem𝑥
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Covering Radius
Definition: Covering radius of with respect to .
ℤ22 (𝐾−𝑥 )+𝑥𝑥Must scale by factor about to hit .
Therefore .
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
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Covering Radius
Definition: Covering radius of with respect to .
ℤ2𝐾 contains a fundamental domain
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Covering Radius
Definition: Further Properties:
ℤ2𝐾
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Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
𝐾𝐾 + 𝑡
ℤ2
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𝐾𝐾 + 𝑡
Flatness Theorem
Theorem: For a convex body and lattice :
By homogeneity of and .
ℤ2
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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)
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Flatness Theorem
Theorem [Banaszczyk `93]: For a lattice in :
𝐿𝑥𝜇
is max distance to
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Flatness Theorem
Theorem [Banaszczyk `96]: For a symmetric convex body :
Bound Improvements:Khinchine `48: Babai `86: Kannan-Lovasz `88: Banaszczyk `93: Banaszczyk `96:
ℤ2𝐾
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Lower Bounds: Random Lattices
Theorem: For a convex body and ``random’’ lattice in :
(Minkowski Hlawka + Volume Product Bound)
ℤ2𝐾
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Symmetric convex body (.
Gauge function:
0
𝑥𝐾 𝑠𝐾
Norms and Convex Bodies
-
1. (triangle inequality) 2. (homogeneity)3. (symmetry)
is unit ball of
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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
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Gauge function:
Triangle Inequality:
Want to show
By definition . May assume Need to show .
Norms and Convex Bodies
convex combination
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asymmetric norm in . Unit ball .
𝑥
𝐾
Norms and Convex Bodies
0
is convex: Take
𝑦-
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𝐾
Symmetric convex body and lattice in .If , such that .
Minkowski’s Convex Body Theorem
0
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𝐾
Symmetric convex body and lattice in .If , such that .
Minkowski’s Convex Body Theorem
0 𝑦
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𝐾
Pf: basis for with parallelepiped .
Minkowski’s Convex Body Theorem
0𝑏1
𝑏22𝑏2
2𝑏1𝑃
2𝑃
![Page 67: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/67.jpg)
𝐾
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowski’s Convex Body Theorem
0
2𝑃
![Page 68: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/68.jpg)
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
![Page 69: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/69.jpg)
Pf: basis for . Let be parallelepiped. Shift tiles intersecting into using .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
![Page 70: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/70.jpg)
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowski’s Convex Body Theorem
0𝐾
2𝑃𝑐
![Page 71: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/71.jpg)
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
+
+ 𝑐
![Page 72: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/72.jpg)
Here (a) , (b) , (c)
Minkowski’s Convex Body Theorem
0𝐾
2𝑃
+
+ 𝑐
![Page 73: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/73.jpg)
Then (by symmetry of ) and .
Minkowski’s Convex Body Theorem
0𝐾
2𝑃𝑐
+
+
2(𝑦−𝑥)
![Page 74: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/74.jpg)
Then (by symmetry of ) and .
Minkowski’s Convex Body Theorem
𝐾0
2(𝑦−𝑥)
2𝐾
![Page 75: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/75.jpg)
So and
Minkowski’s Convex Body Theorem
𝐾0𝑦−𝑥
![Page 76: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/76.jpg)
Symmetric convex body and lattice in . Successive Minima
0𝐾
![Page 77: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/77.jpg)
𝜆1𝐾- 𝑦 1
Symmetric convex body and lattice in . Successive Minima
0
![Page 78: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/78.jpg)
𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in . Successive Minima
𝑦 2𝜆2𝐾
-
0
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𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in .Minkowski’s First Theorem
0
![Page 80: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/80.jpg)
𝑠𝐾
Symmetric convex body and lattice in .Pf: Let . For
Minkowski’s First Theorem
0
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𝑦
Symmetric convex body and lattice in .Pf: By Minkowski’s convex body theorem, .
Since this holds as , .
Minkowski’s First Theorem
0𝑠𝐾
![Page 82: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/82.jpg)
𝑦 2𝜆2𝐾
-
𝑦 1-𝜆1𝐾
Symmetric convex body and lattice in .Minkowski’s Second Theorem
0
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Theorem [Kannan-Lovasz 88]:
For symmetric becomes
Covering Radius vs Successive Minima
“Naïve” Babai rounding
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For a closed convex set and , there exists such that
Separator Theorem
Can use where is the closest point in to .
𝑝𝐾𝑦
𝑥∗
![Page 85: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/85.jpg)
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∗
![Page 86: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/86.jpg)
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)
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Theorem: compact convex with in rel. int., then .In particular .
Polar Bodies and Dual NormsDef:
𝐾0
![Page 88: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/88.jpg)
Theorem: compact convex with in rel. int., then .In particular .
Polar Bodies and Dual NormsDef:
Pf: Easy to check that.
Hence .
Must show .
𝐾0
![Page 89: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/89.jpg)
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
![Page 90: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/90.jpg)
Then and therefore .
Theorem: compact convex with in rel. int., .In particular .
𝑝𝐾
Polar Bodies and Dual Norms
𝑦0
Def:
![Page 91: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/91.jpg)
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
![Page 92: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/92.jpg)
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
![Page 93: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/93.jpg)
Theorem [Kannan-Lovasz `88] : For a symmetric convex body and lattice
Pf: By Minkowski’s first theorem
(Bourgain-Milman)
Minkowski Transference
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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
![Page 95: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/95.jpg)
𝐾𝐾 + 𝑡
Flatness Theorem
We will follow proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :
ℤ2
![Page 96: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/96.jpg)
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
![Page 97: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/97.jpg)
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 )
![Page 98: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/98.jpg)
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
![Page 99: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/99.jpg)
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
![Page 100: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/100.jpg)
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 .
![Page 101: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/101.jpg)
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
![Page 102: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/102.jpg)
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 .
![Page 103: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/103.jpg)
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
![Page 104: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/104.jpg)
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
![Page 105: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/105.jpg)
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 .
![Page 106: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/106.jpg)
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
![Page 107: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/107.jpg)
Flatness Theorem
Theorem: For a convex body and lattice :
For , we have and .
By the Minkowski transference
![Page 108: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/108.jpg)
Flatness Theorem
Theorem: For a convex body and lattice :
By the Minkowski transference
By inclusion .Hence
![Page 109: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/109.jpg)
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
![Page 110: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/110.jpg)
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
![Page 111: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/111.jpg)
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
![Page 112: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/112.jpg)
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 .
![Page 113: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/113.jpg)
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 .
![Page 114: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/114.jpg)
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 .
![Page 115: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/115.jpg)
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 .
![Page 116: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/116.jpg)
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 .
![Page 117: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/117.jpg)
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 .
![Page 118: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/118.jpg)
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 .
![Page 119: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/119.jpg)
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 .
![Page 120: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/120.jpg)
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.
![Page 121: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/121.jpg)
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
.
![Page 122: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/122.jpg)
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
![Page 123: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/123.jpg)
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 , .
![Page 124: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/124.jpg)
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
![Page 125: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/125.jpg)
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 .
![Page 126: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/126.jpg)
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.
![Page 127: Transference Theorems in the Geometry of Numbers](https://reader036.fdocuments.net/reader036/viewer/2022081517/56816864550346895ddeb8dd/html5/thumbnails/127.jpg)
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.
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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: +
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Gaussian Heuristic
Lemma: For a convex body and lattice , and , then for any
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𝐾 +𝑥
Gaussian HeuristicWant to bound .By shifting may assume .
𝐿
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𝐾
Gaussian HeuristicSince covers space, exists fundamental domain .
𝐹𝐿
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𝐿
𝐾
Gaussian HeuristicPlace around each point in .
𝐹
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𝐿
𝐾
Gaussian HeuristicPlace around each point in .
Hence
𝐹
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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 , .
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𝐿
12 𝐾
Subspace FlatnessHave .
𝐾
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𝐿
Subspace FlatnessHence can find projection such that
is small.
𝐾
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𝐿
𝐾
Subspace FlatnessCan find projection such that
is small.
𝑊
𝜋𝑊 (𝐾 )
𝜋𝑊 (𝐿)
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𝐿
𝐾
Subspace FlatnessCorresponds to small number of shifts
of intersecting .
𝑊
𝜋𝑊 (𝐾 )
𝜋𝑊 (𝐿)
+ +…
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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
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Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
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Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Lower bound valid for all . Given lower bound is polytime computable.
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Subspace Flatness for norm
Conjecture [Kannan-Lovasz `88, D `12]:
Promise problem: -coGapCRP (Covering Radius Problem) where
YES instances:
No instances:
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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!)