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Transference Theorems in the Geometry of Numbers. Daniel Dadush New York University EPIT 2013. Convex Bodies. Convex body . (convex, full dimensional and bounded). Convexity: Line between and in . Equivalently . Non convex set. Integer Programming Problem (IP). Input: - PowerPoint PPT Presentation
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Transference Theorems in the Geometry of Numbers
Daniel DadushNew York University
EPIT 2013
Convex body . (convex, full dimensional and bounded).
Convex Bodies
πΎπ₯
π¦
Non convex set.
Convexity: Line between and in .Equivalently
π₯π¦
πΎ 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)
β€ππΎ 2
convex set
πΎ 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)
β€ππΎ 2
convex set
Input: : Invertible Transformation Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
π΅πΎ1
π΅β€π
π΅πΎ 2
Input: Remark: can be restricted to any lattice .
Integer Programming Problem (IP)
πΎ 1
πΏπΎ 2
πΎ 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
β€2πΎ 2
πΎ 1
Examples
β€2
If a convex set very ``fatββ, then it will always contain a lattice point.
βHidden cubeβ
πΎ 1
Examples
β€2
If a convex set very ``fatββ, then it will always contain a lattice point.
Examples
β€2
Volume does NOT guarantee lattice points (in contrast with Minkowskiβs theorem).
Infinite band
Examples
β€2
However, lattice point free sets must be ``flatββ in some direction.
Lattice Width
β€2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
π₯1=1 π₯1=8π₯1=4 β¦
Lattice Width
β€2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
π₯2=1π₯2=2
π₯2=3
π₯2=4
Lattice Width
β€2
For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
Note: axis parallel hyperperplanes do NOT suffice.
π₯1βπ₯2=1 π₯1βπ₯2= 4
Lattice Width
β€2
Why is this useful? IP feasible regions: hyperplane decomposition enables reduction into dimensional sub-IPs.
# intersections # subproblems
π₯2=1π₯2=2
π₯2=3
π₯2=4
πΎ 2
πΎ 1
subproblems
subproblems
Lattice Width
β€2
Why is this useful? # intersections # subproblems
If # intersections is small, can solve IP via recursion.
π₯2=1π₯2=2
π₯2=3
π₯2=4
πΎ 2
πΎ 1
subproblems
subproblems
Lattice Width
Integer Hyperplane : Hyperplane where
Fact: is an integer hyperplane , ( called primitive if )
β€2
π₯1βπ₯2=1 π₯1βπ₯2= 4
π» (1 ,β1)π
Lattice Width
Hyperplane Decomposition of : For
(parallel hyperplanes)
β€2
π₯1=1 π₯1=8
(1 ,0)π
β¦
Lattice Width
Hyperplane Decomposition of : For
(parallel hyperplanes)
Note: If is not primitive, decomposition is finer than necessary.
β€2
2 π₯1=2 2 π₯1=16
(2 ,0)2π
2 π₯1=5 2 π₯1=9β¦β¦
πΎ
Lattice Width
How many intersections with ? (parallel hyperplanes) # INTs }| + 1 (tight within +2)
β€2
π₯1=1 π₯1=8
(1 ,0)ππ₯πππ₯
π₯πππ
2 3 4 6 71.9
7.2
Lattice Width
Width Norm of : for any Lattice Width: width
β€2
πΎ π¦widthπΎ ( π¦ )=1.2
Kinchineβs 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:
πΎ
Properties of
Width Norm of : for any
π π΅2π
π‘
π π΅2π 0
πΎ βπΎ
2π π΅2π
Convex & Centrally Symmetric 2π π΅2
π
πΎ
Properties of
Width Norm of : for any Bounds:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Bounds:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Symmetry: By symmetry of
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Symmetry: Therefore
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Homogeneity: For (Trivial)
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Triangle Inequality:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Triangle Inequality:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Triangle Inequality:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
πΎ
Properties of
Width Norm of : for any Triangle Inequality:
π π΅2π
π‘
π π΅2π 0
2π π΅2π
2π π΅2π
Convex & Centrally Symmetric
πΎ βπΎ
Properties of
is invariant under translations of .
πΎ
π¦
πππ πππ₯π€πππ‘ hπΎ (π¦ )=πππ₯βπππ
Properties of
is invariant under translations of .
πΎ
π¦
πππ πππ₯
πΎ + π‘
π¦
πππ+ β¨π¦ , π‘ β© +
Properties of
is invariant under translations of .Also follows since .(width only looks at differences between vectors of .
πΎ
π¦
πππ πππ₯
πΎ + π‘
π¦
πππ+ β¨π¦ , π‘ β© +
Kinchineβs 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 , .
Kinchineβs 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 , .
Kinchineβs 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 )
Kinchineβs 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 , .
Kinchineβs 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.
ππ1
ππ20
No interior lattice points.
π
Lower Bound: Simplex
Bound cannot be improved to .
Pf: For , then
ππ1
ππ20π β€2
Flatness Theorem
Theorem*: For a convex body and lattice , if such that , then .
By shift invariance of .
πΎπΎ + π‘
β€2
Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
πΎπΎ + π‘
β€2
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎ
Covering Radius
Definition: Covering radius of with respect to .
β€2
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎ Condition from
Flatness Theorem
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎCondition fromFlatness Theoremπ₯
Covering Radius
Definition: Covering radius of with respect to .
β€22 (πΎβπ₯ )+π₯π₯Must scale by factor about to hit .
Therefore .
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎ contains a fundamental domain
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎ contains a fundamental domain
Covering Radius
Definition: Covering radius of with respect to .
β€2πΎ contains a fundamental domain
Covering Radius
Definition: Further Properties:
β€2πΎ
Flatness Theorem
Theorem**: For a convex body and lattice , either1) , or2) .
πΎπΎ + π‘
β€2
πΎπΎ + π‘
Flatness Theorem
Theorem: For a convex body and lattice :
By homogeneity of and .
β€2
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 :
πΏπ₯π
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:
β€2πΎ
Lower Bounds: Random Lattices
Theorem: For a convex body and ``randomββ lattice in :
(Minkowski Hlawka + Volume Product Bound)
β€2πΎ
Symmetric convex body (.
Gauge function:
0
π₯πΎ π πΎ
Norms and Convex Bodies
-
1. (triangle inequality) 2. (homogeneity)3. (symmetry)
is unit ball of
Convex body containing origin in its interior.
Gauge function: π₯πΎ π πΎ
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 .
π₯
πΎ
Norms and Convex Bodies
0
is convex: Take
π¦-
πΎ
Symmetric convex body and lattice in .If , such that .
Minkowskiβs Convex Body Theorem
0
πΎ
Symmetric convex body and lattice in .If , such that .
Minkowskiβs Convex Body Theorem
0 π¦
πΎ
Pf: basis for with parallelepiped .
Minkowskiβs Convex Body Theorem
0π1
π22π2
2π1π
2π
πΎ
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowskiβs Convex Body Theorem
0
2π
Pf: basis for . Let be parallelepiped. Tile space with using .
Minkowskiβs Convex Body Theorem
0πΎ
2π
Pf: basis for . Let be parallelepiped. Shift tiles intersecting into using .
Minkowskiβs Convex Body Theorem
0πΎ
2π
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowskiβs Convex Body Theorem
0πΎ
2ππ
Pf: basis for . Let be parallelepiped. Since , must have intersections.
Minkowskiβs Convex Body Theorem
0πΎ
2π
+
+ π
Here (a) , (b) , (c)
Minkowskiβs Convex Body Theorem
0πΎ
2π
+
+ π
Then (by symmetry of ) and .
Minkowskiβs Convex Body Theorem
0πΎ
2ππ
+
+
2(π¦βπ₯)
Then (by symmetry of ) and .
Minkowskiβs Convex Body Theorem
πΎ0
2(π¦βπ₯)
2πΎ
So and
Minkowskiβs Convex Body Theorem
πΎ0π¦βπ₯
Symmetric convex body and lattice in . Successive Minima
0πΎ
π1πΎ- π¦ 1
Symmetric convex body and lattice in . Successive Minima
0
π¦ 1-π1πΎ
Symmetric convex body and lattice in . Successive Minima
π¦ 2π2πΎ
-
0
π¦ 1-π1πΎ
Symmetric convex body and lattice in .Minkowskiβs First Theorem
0
π πΎ
Symmetric convex body and lattice in .Pf: Let . For
Minkowskiβs First Theorem
0
π¦
Symmetric convex body and lattice in .Pf: By Minkowskiβs convex body theorem, .
Since this holds as , .
Minkowskiβs First Theorem
0π πΎ
π¦ 2π2πΎ
-
π¦ 1-π1πΎ
Symmetric convex body and lattice in .Minkowskiβs Second Theorem
0
Theorem [Kannan-Lovasz 88]:
For symmetric becomes
Covering Radius vs Successive Minima
βNaΓ―veβ Babai rounding
For a closed convex set and , there exists such that
Separator Theorem
Can use where is the closest point in to .
ππΎπ¦
π₯β
For a closed convex set and , there exists such that
ππΎ
Separator Theorem
π¦
If not separator can get closer to on line segment .
π§π₯ββπ₯β
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:
πΎ0
Theorem: compact convex with in rel. int., then .In particular .
Polar Bodies and Dual NormsDef:
Pf: Easy to check that.
Hence .
Must show .
πΎ0
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
ππΎπ¦0
Then and therefore .
Theorem: compact convex with in rel. int., .In particular .
ππΎ
Polar Bodies and Dual Norms
π¦0
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
Banaszczykβs 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 Minkowskiβs 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
πΎπΎ + π‘
Flatness Theorem
We will follow proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :
β€2
Flatness Theorem
We will follow proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :
Proof of lower bound:
πΎπΎ + π‘
β€2
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
πΎβ€2
π1π π§ 1
π π§ 2
πΎ
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
β€2ππΎπ1
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 .
πΎ
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 .
β€2
π1
π2π₯π₯
π¦
π 2
πΎ
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 .
πΎ
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. .
β€2
πΌπ1π₯π₯
π¦
π 2
πΎ
οΏ½ΜοΏ½
π1
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
πΎ
πΎπΎ
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
β€2
πΎπΎ
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
β€2
πΎ
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
β€2
πΎ
: r=2
0
π»={0 }
β€2
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 .
πΎ
β€2
: 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 .
πΎ
β€2
: 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 .
πΎ
β€2
π»={( 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 .
πΎ
β€2
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 .
πΎ
β€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 .
πΎ
If , is a hyperplane.Forcing corresponds to Classical Flatness Theorem.
β€2
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 .
πΎ
β€2
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 Minkowskiβs 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 ``largeββ volume (i.e. relative to determinant) in every projection implies ``alwaysββ contains lattice points.
In this sense, we get a generalization of Minkowskiβs 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 Minkowskiβs first theorem
.
Brunn-Minkowski: +
Gaussian Heuristic
Lemma: For a convex body and lattice , and , then for any
πΎ +π₯
Gaussian HeuristicWant to bound .By shifting may assume .
πΏ
πΎ
Gaussian HeuristicSince covers space, exists fundamental domain .
πΉπΏ
πΏ
πΎ
Gaussian HeuristicPlace around each point in .
πΉ
πΏ
πΎ
Gaussian HeuristicPlace around each point in .
Hence
πΉ
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 , .
πΏ
12 πΎ
Subspace FlatnessHave .
πΎ
πΏ
Subspace FlatnessHence can find projection such that
is small.
πΎ
πΏ
πΎ
Subspace FlatnessCan find projection such that
is small.
π
ππ (πΎ )
ππ (πΏ)
πΏ
πΎ
Subspace FlatnessCorresponds to small number of shifts
of intersecting .
π
ππ (πΎ )
ππ (πΏ)
+ +β¦
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!)
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