论文标题
Q-Moment措施和应用：通过最佳运输的新方法
q-Moment Measures and Applications: A New Approach via Optimal Transport
论文作者
论文摘要
2017年，Bo'az Klartag获得了有关椭圆类仿射半球存在差异几何形状的新结果。在他的方法中，一个表面与每个凸功能$φ$：r^n $ \ rightarrow $（0， +$ \ infty $）以及表面成为仿射半球的条件涉及$φ$的2级量度（Q-Moment的特定案例，即$ \ nabla $ $ \ nabla $φ或＃ （$φ$^{ - （n+q）}）用于q> 0）。在克拉塔格（Klartag）的论文中，通过一种差异方法研究了Q-Moment措施，该方法需要最大程度地减少凸函数之间的功能，该功能是使用Borell-Borell-Brascamp-lieb不平等的研究。在本文中，我们通过最佳运输方法攻击了相同的问题，因为凸函数$φ$是坎多维奇的潜力（就像以前的论文中已经完成了时刻量度一样）。在这种新方法中，变异问题成为局部功能和概率度量中的运输成本的最小化，而优化器证明是$ρ$ = $ = $φ$^{ - （n+q）}的形式。
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)}.
