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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

Filippo Santambrogio
Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France


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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