Ehrhart polynomial and Successive Minima

Abstract

We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space ${\mathbb{R}}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related by their geometric and arithmetic mean. We also show that the roots of lattice $n$-polytopes with or without interior lattice points differ essentially. Furthermore, we study the structure of the roots in the planar case. Here it turns out that their distribution reflects basic properties of lattice polygons.

References in corpus (1)

• Ehrhart polynomials and stringy Betti numbers

Cited by in corpus (12)

• The $L_p$-Minkowski problem for $-n<p<1$
• Chord Measures in Integral Geometry and Their Minkowski Problems
• Roots of Ehrhart polynomials arising from graphs
• The Orlicz version of the $L_p$ Minkowski problem on $S^{n-1}$ for $-n<p<0$
• The log-Brunn-Minkowski inequality in ${\mathbb{R}}^3$
• Fourier transforms of polytopes, solid angle sums, and discrete volume
• Interpolating between volume and lattice point enumerator with successive minima
• Subspace concentration of dual curvature measures
• Notes on the roots of Ehrhart polynomials
• On extensions of Minkowski's theorem on successive minima
• Notes on lattice points of zonotopes and lattice-face polytopes
• Minkowski's successive minima in convex and discrete geometry