Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020 pp. 67–91.

Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem

Authors: Alexander V. Kolesnikov (1)
Author institution:(1) National Research University Higher School of Economics, Russian Federation

Summary:

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 $S^{n-1}$: $\frac{1}{n} \operatorname{Ent}(\nu) \ge K({\sigma}, \nu)$. 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.

2010 Math. Subj. Class. 52A40, 90C08.

Keywords: Convex bodies, optimal transportation, Kantorovich duality, log-Minkowski problem, Kähler–Einstein equation

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