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

We study the transportation problem on the unit sphere Sⁿ⁻¹ for symmetric probability measures and the cost function c(x,y) = log 1/ 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: 1/n Ent(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.