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In the field of optimal transport theory, an optimal map is known to be agradient map of a potential function satisfying cost-convexity. In this paper,the Jacobian determinant of a gradient map is shown to be log-concave withrespect to a convex combination of the potential functions when the underlyingmanifold is the sphere and the cost function is the distance squared. The proofuses the non-negative cross-curvature property of the sphere recentlyestablished by Kim and McCann, and Figalli and Rifford. As an application tostatistics, a new family of probability densities on the sphere is defined interms of cost-convex functions. The log-concave property of the likelihoodfunction follows from the inequality.

A Jacobian inequality for gradient maps on the sphere and its application to directional statistics