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

Moscow Mathematical Journal. 2020. Vol. 20. No. 1. P. 67–91.

Kolesnikov A.

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩.
We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−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 σ on Sn−1:
1nEnt(ν)≥K(σ,ν). It is shown that there exists a remarkable similarity between our results and the theory of the K{ä}hler-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.

Keywords: сфера, Optimal transportation, transportation inequalities, транспортные неравенства, оптимальная транспортировка, log Brunn-Minkowski inequality, логарифмическое неравенство Брунна-Минковского, sphere

Publication based on the results of: Стохастические алгоритмы и статистический анализ многомерных данных (2018)