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Ansgar Freyer

Publications and source records attributed to Ansgar Freyer.

14 recordsLinked to original sources
arXiv2026

Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it is related to the classical flatness constant as well as a conjectural dual version of Minkowski's convex body theorem due to Makai. Moreover, it is shown that Flt(2, 1) = 3, i.e., any planar convex body with at most one interior point has lattice width at most three. This leads to an isominwidth inequality for the lattice point enumerator of planar convex bodies.

math.MG↗
arXiv2026

Minimal covering bodies and Brunn-Minkowski type inequalities for the covering radius

Inclusion minimal convex bodies $K$ with the property that the integer translates of $K$ cover the space are studied. Such bodies are referred to as minimal covering bodies and it is shown that, while they are not necessarily tiles, they are polytopes with at least $2d$ facets, if $d$ is the dimension of $K$. Moreover, minimal covering bodies are related to covering properties of Minkowski combinations of convex bodies. Two sharp Brunn Minkowski type inequalities are established for the covering radius of the Minkowski sum of planar convex bodies.

math.MG↗
arXiv2025

Exponential valuations on lattice polygons valued at formal power series

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The classification is done in terms of formal power series that satisfy certain functional equations. We align our classification with the decomposition into so-called dilative components.

math.MG↗
arXiv2025

Pal's isominwidth problem in the hyperbolic space

The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most π/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π/2. In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle.

math.MG↗
arXiv2025

The canonical form, scissors congruence and adjoint degrees of polytopes

We study the canonical form $Ω$ as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial $\operatorname{adj}_P$ - as an important invariant in this context. More precisely, for a polytope $P$ we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of $P$ is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form $Ω_0$ and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if $P$ has positive degree drop. Using it we can prove that zonotopes can be characterized as the $d$-polytopes that have maximal possible degree drop $d-1$. We obtain a decomposition formula for $Ω_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $Ω_s$ that can distinguish higher values of the degree drop.

math.CO↗
arXiv2025

Exponential valuations on lattice polygons

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.

math.NT↗
arXiv2024

The isominwidth problem on the 2-sphere

Pál's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $\tfrac π2$. If the width is greater than $\tfrac π2$, the regular triangle no longer minimizes the area at fixed minimal width. We show that the minimizers are instead given by the polar sets of spherical Reuleaux triangles. Moreover, stability versions of the two spherical inequalities are obtained.

math.MG↗
arXiv2024

Lattice Reduced and Complete Convex Bodies

The purpose of this paper is to study convex bodies $C$ for which there exists no convex body $C^\prime\subsetneq C$ of the same lattice width. Such bodies shall be called ``lattice reduced'', and they occur naturally in the study of the flatness constant in integer programming, as well as other problems related to lattice width. We show that any simplex that realizes the flatness constant must be lattice reduced and prove structural properties of general lattice reduced convex bodies: they are polytopes with at most $2^{d+1}-2$ vertices and their lattice width is attained by at least $Ω(\log d)$ independent directions. Strongly related to lattice reduced bodies are the ``lattice complete bodies'', which are convex bodies $C$ for which there exists no $C^\prime\supsetneq C$ such that $C^\prime$ has the same lattice diameter as $C$. Similar structural results are obtained for lattice complete bodies. Moreover, various construction methods for lattice reduced and complete convex bodies are presented.

math.MG↗
arXiv2024

Unimodular Valuations beyond Ehrhart

A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behaviour with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

math.MG↗
arXiv2024

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 $c\,n^2\,ω(n)/\log(n)$, where $c$ is an absolute constant and $ω(n)$ is the flatness constant. Due to the recent best known upper bound on $ω(n)$ 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↗
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↗
arXiv2022

Interpolating between volume and lattice point enumerator with successive minima

We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke's shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski's upper bound on the volume in terms of the successive minima.

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↗