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## 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)=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.