arXiv 2008.093622020-08-21math.AP

q-矩测度及其应用：基于最优输运的一种新方法

q-Moment Measures and Applications: A New Approach via Optimal Transport

Huynh Khanh, Filippo Santambrogio

摘要：

2017年，Bo'az Klartag在微分几何中获得了关于椭圆型仿射半球存在性的新结果。在他的方法中，曲面与每个凸函数$Φ$相关联：R^n $\rightarrow$（0，+ $\infty$），曲面成为仿射半球的条件涉及$Φ$的2-矩测度（q-矩测度的一种特殊情况，即（$\nabla$ $Φ$）\#（$Φ$ ^{--（n+q）}）（q > 0）形式的测度）。在Klartag的论文中，q-矩测度是通过一种变分方法来研究的，该方法要求最小化凸函数中的一个泛函，并使用Borell-Brascamp-Lieb不等式来研究。在本文中，我们攻击相同的问题，通过一个最优输运的方法，因为凸函数$Φ$是一个康托洛维奇潜力（如已经做了矩措施在以前的文件）。在这种新方法中的变分问题变成了概率测度之间的局部泛函和传输成本的最小化，并且优化器的形式为$ρ$ = $Φ$ ^{--（n+q）}。

英文摘要：

In 2017, Bo'az Klartag obtained a new result in differential geometry on the existence of affine hemisphere of elliptic type. In his approach, a surface is associated with every a convex function $Φ$ : R^n $\rightarrow$ (0, +$\infty$) and the condition for the surface to be an affine hemisphere involves the 2-moment measure of $Φ$ (a particular case of q-moment measures, i.e measures of the form ($\nabla$$Φ$) \# ($Φ$^{--(n+q)}) for q > 0). In Klartag's paper, q-moment measures are studied through a variational method requiring to minimize a functional among convex functions, which is studied using the Borell-Brascamp-Lieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function $Φ$ is a Kantorovich potential (as already done for moment measures in a previous paper). The variational problem in this new approach becomes the minimization of a local functional and a transport cost among probability measures and the optimizer turns out to be of the form $ρ$ = $Φ$^{--(n+q)}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文