Motivated by the discrete logarithmic Minkowski problem we study for a given matrix U^n m its cone-volume set C ₂ₕ (U) consisting of all the cone-volume vectors of polytopes P (U, b) =\ xⁿ: U^ x b\, bⁿ ₀. We will show that C ₂ₕ (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 ₒ₂₂ (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 ₒ₂₂ (U) and C ₂ₕ (U), also offer a new geometric point of view to the discrete logarithmic Minkowski problem.
Baumbach et al. (Wed,) studied this question.