Discrete Mixed Volumes and Alexandrov–fenchel-type Inequalities
Year: 2026
Type: preprint
Abstract: Motivated by the parallels between volume and lattice point counting, we study the mixed version of Ehrhart theory, computing the number of lattice points in Minkowski sums of lattice polytopes. This quantity is a polynomial in the scalars, and the coefficients in the binomial basis are given by the discrete mixed volumes, which generalize mixed volumes in the case of lattice polytopes. We prove a general asymptotic Alexandrov-Fenchel-type inequality for the discrete mixed volumes of lattice polytopes, and an exact inequality for coordinate simplices. To prove the exact inequality, we use the exceptional Hirzebruch-Riemann-Roch theorem of Berget-Eur-Spink-Tseng on the permutohedral variety to express the discrete mixed volume as a classical mixed volume. Additionally, we give a matroidal interpretation of this result, showing that these quantities are exactly the coefficients of the characteristic polynomial of the associated cotransversal matroid, recovering their log-concavity.
Source: HAL
Author: Hakavuori, Joel
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220303705