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.
Categories: math.MG