On polynomial inequalities for cone-volumes of polytopes

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix Un× m its cone-volume set Ccv(U) consisting of all the cone-volume vectors of polytopes P(U,b)=\ xⁿ : U^ x≤ b\, bⁿ≥ 0. We will show that Ccv(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 Pscc(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, Pscc(U) and Ccv(U), also offer a new geometric point of view to the discrete logarithmic Minkowski problem.