Journal of Convex Analysis 28 (2021), No. 4, 1033--1052
Copyright Heldermann Verlag 2021
q-Moment Measures and Applications: a New Approach via Optimal Transport
Huynh Khanh
Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
khanh.edu02@gmail.com
Filippo Santambrogio
Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France
santambrogio@math.univ-lyon1.fr
[Abstract-pdf]
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 convex function $\varphi\colon {\mathbb R}^n \to (0, +\infty)$ and the condition for the surface to be an affine hemisphere involves the 2-moment measure of $\varphi$ (a particular case of $q$-moment measures, i.e measures of the form ${(\nabla \varphi)_\# }{\varphi^{-({n + q})}}$ 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 achieved using the Borell-Brascamp-Lieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function $\varphi$ 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 $\varrho$ and the optimizer $\varrho_{\rm {opt}}$ turns out to be of the form $\varrho_{\rm {opt}} = \varphi^{-(n + q)}$.
Keywords: Affine spheres, convex functions, Wasserstein spaces.
MSC: 49J45, 14R05, 35J96.
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