Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

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


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

Contents   Full-Text PDF