On polynomial inequalities for cone-volumes of polytopes

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 $C_{\mathtt{cv}}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{x\in {\mathbb{R}}^n:U^{⊺}x\le b\}$, $b\in {\mathbb{R}}_{\ge 0}^n$. We will show that $C_{\mathtt{cv}}(U)$ 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 $P_{\mathtt{scc}}(U)$ 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, $P_{\mathtt{scc}}(U)$ and $C_{\mathtt{cv}}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.