• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

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

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

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.