On the covering radius of lattice polytopes and its relation to - PowerPoint PPT Presentation

On the covering radius of lattice polytopes and its relation to view-obstructions and densities of lattice arrangements Matthias Schymura (n´ e Henze) Freie Universit¨ at Berlin based on joint work with Bernardo Gonz´ alez Merino Romanos-Diogenes Malikiosis Technische Universit¨ at M¨ unchen Technische Universit¨ at Berlin December 12, 2016 Einstein Workshop on Lattice Polytopes Freie Universit¨ at Berlin

Lattices of Convex Bodies Definition For a convex body K in R n and a lattice Λ = A Z n , A ∈ GL n ( R ), we say that � K + Λ = ( K + z ) z ∈ Λ is a lattice of translates of K . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 2 / 21

Lattices of Convex Bodies Definition For a convex body K in R n and a lattice Λ = A Z n , A ∈ GL n ( R ), we say that � K + Λ = ( K + z ) z ∈ Λ is a lattice of translates of K . + Z n = Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 2 / 21

Lattices of Convex Bodies Definition For a convex body K in R n and a lattice Λ = A Z n , A ∈ GL n ( R ), we say that � K + Λ = ( K + z ) z ∈ Λ is a lattice of translates of K . Z n = + Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 2 / 21

Lattices of Convex Bodies Definition For a convex body K in R n and a lattice Λ = A Z n , A ∈ GL n ( R ), we say that � K + Λ = ( K + z ) z ∈ Λ is a lattice of translates of K . = + Z n Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 2 / 21

Lattices of Convex Bodies Definition For a convex body K in R n and a lattice Λ = A Z n , A ∈ GL n ( R ), we say that � K + Λ = ( K + z ) z ∈ Λ is a lattice of translates of K . = + Z n Definition The lattice of translates K + Λ is a lattice covering if K + Λ = R n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 2 / 21

Covering Radius Definition The covering radius of K ⊆ R n with respect to a lattice Λ is defined as µ ( K , Λ) = min { µ > 0 : µ K + Λ = R n } . We abbreviate µ ( K ) = µ ( K , Z n ). Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 3 / 21

Covering Radius Definition The covering radius of K ⊆ R n with respect to a lattice Λ is defined as µ ( K , Λ) = min { µ > 0 : µ K + Λ = R n } . We abbreviate µ ( K ) = µ ( K , Z n ). Appearances in the literature: Coin Exchange Problem of Frobenius (Kannan ’92) Transference Theorems, Diophantine Approximation (Kannan & Lov´ asz ’88) Flatness Theorem (Khinchin ’54; Lagarias, Lenstra & Schnorr ’90; Banaszczyk ’96) Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 3 / 21

Covering Radius Definition The covering radius of K ⊆ R n with respect to a lattice Λ is defined as µ ( K , Λ) = min { µ > 0 : µ K + Λ = R n } . We abbreviate µ ( K ) = µ ( K , Z n ). Appearances in the literature: Coin Exchange Problem of Frobenius (Kannan ’92) Transference Theorems, Diophantine Approximation (Kannan & Lov´ asz ’88) Flatness Theorem (Khinchin ’54; Lagarias, Lenstra & Schnorr ’90; Banaszczyk ’96) Computationally difficult parameter: Kannan ’93: Polynomial-time algorithm to compute µ ( P , Λ) for rational polytopes P in fixed dimension; triple-exponential in the dimension. Haviv & Regev ’06: It is Π 2 -hard to approximate µ ( B n p , Λ) to within a factor c p > 0 for all sufficiently large p ≥ 1. (Conjecture) Deciding µ ( B n 2 , Λ) ≤ µ is NP-hard. (Guruswami et al. ’05) Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 3 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). µ 2 ( K ) = 4 3 Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). µ 2 ( K ) = 4 3 Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). µ 2 ( K ) = 4 3 and µ 1 ( K ) = 1 Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). µ 1 ( K ) ≤ µ 2 ( K ) ≤ . . . ≤ µ n ( K ) = µ ( K ) µ i ( UK ) = µ i ( K ), for 1 ≤ i ≤ n and U ∈ GL n ( Z ) µ i ( rK ) = 1 r µ i ( K ), for 1 ≤ i ≤ n and r > 0 µ i ( AK , A Z n ) = µ i ( K , Z n ), for 1 ≤ i ≤ n and A ∈ GL n ( R ) µ 2 ( K ) = 4 3 and µ 1 ( K ) = 1 Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Covering minima Definition (Kannan & Lov´ asz ’88; G. Fejes T´ oth ’76) The i th covering minimum of K ⊆ R n with respect to a lattice Λ is defined as µ i ( K , Λ) = min { µ > 0 : µ K + Λ intersects every ( n − i )-dim. affine subspace } . We abbreviate µ i ( K ) = µ i ( K , Z n ). µ 1 ( K ) ≤ µ 2 ( K ) ≤ . . . ≤ µ n ( K ) = µ ( K ) µ i ( UK ) = µ i ( K ), for 1 ≤ i ≤ n and U ∈ GL n ( Z ) µ i ( rK ) = 1 r µ i ( K ), for 1 ≤ i ≤ n and r > 0 µ i ( AK , A Z n ) = µ i ( K , Z n ), for 1 ≤ i ≤ n and A ∈ GL n ( R ) µ 2 ( K ) = 4 3 and µ 1 ( K ) = 1 Lemma (Kannan & Lov´ asz ’88) µ i ( K , Λ) = max { µ ( K | L , Λ | L ) : L an i-dimensional subspace } Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 4 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . For S 1 = conv { 0 , e 1 , . . . , e n } , we have µ n ( S 1 ) = n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . For S 1 = conv { 0 , e 1 , . . . , e n } , we have µ n ( S 1 ) = n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . For S 1 = conv { 0 , e 1 , . . . , e n } , we have µ i ( S 1 ) = i for each i = 1 , . . . , n . Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . For S 1 = conv { 0 , e 1 , . . . , e n } , we have µ i ( S 1 ) = i for each i = 1 , . . . , n . For the Euclidean unit ball B n 2 , we have √ i µ i ( B n 2 ) = for each i = 1 , . . . , n . 2 Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

Examples For C n = [ − 1 2 , 1 2 ] n , we have µ i ( C n ) = 1 for each i = 1 , . . . , n . For S 1 = conv { 0 , e 1 , . . . , e n } , we have µ i ( S 1 ) = i for each i = 1 , . . . , n . For the Euclidean unit ball B n 2 , we have √ i µ i ( B n 2 ) = for each i = 1 , . . . , n . 2 Proposition Let P ⊆ R n be a lattice polytope. Then µ i ( P ) ≤ i, for every i = 1 , . . . , n, and if P is a lattice zonotope, then µ i ( P ) ≤ 1 , for every i = 1 , . . . , n. Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 5 / 21

What’s coming? We discuss two problems in which the computation / estimation of covering radii of lattice polytopes plays a crucial role: ➊ Towards a Covering Analog of Minkowski’s 2nd Theorem ➋ Rationally Constrained View-Obstruction Problem Matthias Schymura Covering radii of lattice polytopes Dec 12, 2016 6 / 21