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We consider Hilbert and Funk geometries on a strongly convex domain in theEuclidean space. We show that, with respect to the Lebesgue measure on thedomain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative)weighted Ricci curvature. As one of corollaries, these metric measure spacessatisfy the curvature-dimension condition in the sense of Lott, Sturm andVillani.

Weighted Ricci curvature estimates for Hilbert and Funk geometries