For any finite set of unit normals positively spanning R^n, the cone-volume set C_cv(U) is a path-connected semialgebraic set, and it equals the scaled matroid base polytope only for centrally symmetric parallelepipeds.
The Logarithmic Minkowski Problem
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In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.
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On polynomial inequalities for cone-volumes of polytopes
For any finite set of unit normals positively spanning R^n, the cone-volume set C_cv(U) is a path-connected semialgebraic set, and it equals the scaled matroid base polytope only for centrally symmetric parallelepipeds.