On polynomial inequalities for cone-volumes of polytopes

TL;DR

This paper investigates the geometric properties of cone-volume sets of polytopes related to the discrete logarithmic Minkowski problem, establishing their connectivity and algebraic structure, and introduces a new subspace concentration polytope for measure conditions.

Contribution

It extends the understanding of cone-volume sets to higher dimensions, proves their path-connectedness and semialgebraic nature, and introduces a new geometric polytope related to subspace concentration conditions.

Findings

C_{cv}(U) is a path-connected semialgebraic set.

Introduces the subspace concentration polytope P_{scc}(U).

Provides a new geometric perspective on the discrete logarithmic Minkowski problem.

Abstract

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.