Searcharxiv⌕ Search

arXiv subjects

Martin Henk

Publications and source records attributed to Martin Henk.

At least 19 recordsLinked to original sources
arXiv2026

Packing minima of convex bodies

In 2021, Henk, Schymura and Xue introduced packing minima, associated with a convex body and a lattice, as packing counterparts to the covering minima of Kannan and Lov\'asz. Motivated by conjectures on the volume inequalities for the successive minima, we generalized the definition of the packing minima to the class of all convex bodies that contain the origin in their interior. For these packing minima, we presented several novel volume inequalities and calculated the specific values of the packing minima for several special convex bodies.

math.MG↗
arXiv2025

On polynomial inequalities for cone-volumes of polytopes

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.

math.MG↗
arXiv2023

Sparsity and integrality gap transference bounds for integer programs

We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{cx: x in P, x integer}, where P={x: Ax=b, x nonnegative} is a polyhedron in the standard form determined by an integer mxn matrix A and an integer vector b. The main result of the paper shows that the integrality gap drops exponentially in the size of support of the optimal solutions that correspond to the vertices of the integer hull of the polyhedron P. Additionally, we obtain a new proximity bound that estimates the distance from any point of P to its nearest integer point in P. The proofs make use of the results from the geometry of numbers and convex geometry.

math.OC↗
arXiv2023

Minkowski's successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

math.MG↗
arXiv2023

Polynomial Bounds in Koldobsky's Discrete Slicing Problem

In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace $H\subset\mathbb{R}^n$ with \[ |K\cap\mathbb Z^n| \leq d_n\,|K\cap H\cap \mathbb Z^n|\,\mathrm{vol}(K)^{\frac 1n}. \] In this article we show that $d_n$ is bounded from above by $c\,n^2\,\omega(n)/\log(n)$, where $c$ is an absolute constant and $\omega(n)$ is the flatness constant. Due to the recent best known upper bound on $\omega(n)$ we get a ${c\,n^3\log(n)^2}$ bound on $d_n$. This improves on former bounds which were exponential in the dimension.

math.MG↗
arXiv2023

On subspace concentration for dual curvature measures

We study subspace concentration of dual curvature measures of convex bodies $K$ satisfying $\gamma (-K)\subseteq K$ for some $\gamma \in (0,1]$. We present upper bounds on the subspace concentration depending on $\gamma$, which, in particular, retrieves the known results in the symmetric setting. The proof is based on a unified approach to prove necessary subspace concentration conditions via the divergence theorem.

math.MG↗
arXiv2022

New Bounds for the Integer Carath\'{e}odory Rank

Given a rational pointed $n$-dimensional cone $C$, we study the integer Carath\'{e}odory rank $\operatorname{CR}(C)$ and its asymptotic form $\operatorname{CR^{\rm a}}(C)$, where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for $\operatorname{CR^{\rm a}}(C)$. We also study bounds on $\operatorname{CR}(C)$ in terms of $\Delta$, the maximal absolute $n\times n$ minor of the matrix given in an integral polyhedral representation of $C$. If $\Delta\in\lbrace 1,2\rbrace$, we show $\operatorname{CR}(C) = n$, and prove upper bounds for simplicial cones, improving the best known upper bound on $\operatorname{CR}(C)$ for $\Delta\leq n$.

math.CO↗
arXiv2022

Affine Subspace Concentration Conditions for Centered Polytopes

Recently, K.-Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.

math.MG↗
arXiv2021

On lattice width of lattice-free polyhedra and height of Hilbert bases

We study the lattice width of lattice-free polyhedra given by $\mathbf{A}\mathbf{x}\leq\mathbf{b}$ in terms of $\Delta(\mathbf{A})$, the maximal $n\times n$ minor in absolute value of $\mathbf{A}\in\mathbb{Z}^{m\times n}$. Our main contribution is to link the lattice width of lattice-free polyhedra to the height of Hilbert bases and to the diameter of finite abelian groups. This leads to a bound on the lattice width of lattice-free pyramids which solely depends on $\Delta(\mathbf{A})$ provided a conjecture regarding the height of Hilbert bases holds. Further, we exploit a combination of techniques to obtain novel bounds on the lattice width of simplices. A second part of the paper is devoted to a study of the above mentioned Hilbert basis conjecture. We give a complete characterization of the Hilbert basis if $\Delta(\mathbf{A}) = 2$ which implies the conjecture in that case and prove its validity for simplicial cones.

math.CO↗
arXiv2020

Distance-sparsity transference for vertices of corner polyhedra

We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it gives an exponential (in the size of support of a solution) improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support.

math.OC↗
arXiv2020

Packings, sausages and catastrophes

In this survey we give an overview about some of the main results on parametric densities, a concept which unifies the theory of finite (free) packings and the classical theory of infinite packings.

math.MG↗
arXiv2020

Packing minima and lattice points in convex bodies

Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $Λ$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.

math.MG↗
arXiv2020

Bounds on the lattice point enumerator via slices and projections

Gardner, Gronchi and Zong posed the problem to find a discrete analogue of M. Meyer's inequality bounding the volume of a convex body from below by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question to bound the number of lattice points of a convex body in terms of slices as well as projections.

math.MG↗
arXiv2019

On a discrete John-type theorem

As a discrete counterpart to the classical John theorem on the approximation of (symmetric) $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions $P(A,b)\subset Z^n$ in order to cover (many of) the lattice points inside a convex body by a simple geometric structure. Among others, they proved that there exists a generalized arithmetic progressions $P(A,b)$ such that $P(A,b)\subset K\cap Z^n\subset P(A,O(n)^{3n/2}b)$. Here we show that this bound can be lowered to $n^{O(\ln n)}$ and study some general properties of so called unimodular generalized arithmetic progressions.

math.CO↗
arXiv2018

Distances to Lattice Points in Knapsack Polyhedra

We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower bound.

math.CO↗
arXiv2017

A characterization of dual quermassintegrals and the roots of dual steiner polynomials

For any $I\subset\mathbb{R}$ finite with $0\in I$, we provide a characterization of those tuples $(ω_i)_{i\in I}$ of positive numbers which are dual quermaßintegrals of two star bodies. It turns out that this problem is related to the moment problem. Based on this relation we also get new inequalities for the dual quermaßintegrals. Moreover, the above characterization will be the key tool in order to investigate structural properties of the set of roots of dual Steiner polynomials of star bodies.

math.MG↗