Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = \log \frac{1}{\langle x, y \rangle}$. We calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that $K$ satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure ${\sigma}$ on $S^{n-1}$: $\frac{1}{n} \mathrm{Ent}(\nu) \ge K({\sigma}, \nu)$. It is shown that there exists a remarkable similarity between our results and the theory of the Kähler–Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.