A non-zero even measure on the sphere is the cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition.
Chen, Lp Minkowski problem with not necessarily positive data , Adv
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The Logarithmic Minkowski Problem
A non-zero even measure on the sphere is the cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition.