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

2018/07/18 by Alexander V. Kolesnikov, Kolesnikov, Alexander V. · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1807.07002

openalex publication_date 2018/07/18 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

We study the transportation problem on the unit sphere S^n-1 for\nsymmetric probability measures and the cost function c(x,y) = \log\n\(1)/(\⟨ x, y \⟩).\n We calculate the variation of the corresponding Kantorovich functional K\nand study a naturally associated metric-measure space on S^n-1 endowed with\na Riemannian\n metric generated by the corresponding transportational potential. We\nintroduce a new transportational functional which minimizers are\n solutions to the symmetric log-Minkowski problem and prove that K satisfies\nthe following analog of the Gaussian transportation inequality for the uniform\nprobability measure \σ on S^n-1:\n \(1)/(n) Ent(\ν) \≥ K(\σ, \ν). It is shown that there exists a\nremarkable similarity between our results and the theory of the\nK "ahler-Einstein equation on Euclidean space.\n As a by-product we obtain a new proof of uniqueness of solution to the\nlog-Minkowski problem for the uniform measure.\n

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• On p-Brunn-Minkowski and Brascamp-Lieb inequalities

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