{"title":"Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem","authors":"A. Kolesnikov","doi":"10.17323/1609-4514-2020-20-1-67-91","DOIUrl":null,"url":null,"abstract":"We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = \\log \\frac{1}{\\langle x, y \\rangle}$. \nWe calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ endowed with a Riemannian \nmetric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are \nsolutions to the symmetric log-Minkowski problem and prove that $K$ satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure ${\\sigma}$ on $S^{n-1}$: \n$\\frac{1}{n} Ent(\\nu) \\ge K({\\sigma}, \\nu)$. It is shown that there exists a remarkable similarity between our results and the theory of the K{\\\"a}hler-Einstein equation on Euclidean space. \nAs a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2020-20-1-67-91","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 12
Abstract
We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = \log \frac{1}{\langle x, y \rangle}$.
We calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ 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 ${\sigma}$ on $S^{n-1}$:
$\frac{1}{n} Ent(\nu) \ge K({\sigma}, \nu)$. It is shown that there exists a remarkable similarity between our results and the theory of the K{\"a}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.
期刊介绍:
The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular.
An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.