Abstract

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 φ ⁣:Rn→(0,+∞)\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 φ\varphi (a particular case of qq-moment measures, i.e measures of the form (∇φ)#φ−(n+q){(\nabla \varphi)_\# }{\varphi^{-({n + q})}} for q>0q > 0). In Klartag's paper, qq-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 φ\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 ϱ\varrho and the optimizer ϱopt\varrho_{\rm {opt}} turns out to be of the form ϱopt=φ−(n+q)\varrho_{\rm {opt}} = \varphi^{-(n + q)}.

Author information

Contact details are reproduced from the original publication and may be historical.

Huynh Khanh

Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

khanh.edu02@gmail.com

Keywords

  • Affine spheres
  • convex functions
  • Wasserstein spaces

Mathematics Subject Classification

49J45, 14R05, 35J96

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Suggested citation

H. Khanh, F. Santambrogio. “q-Moment Measures and Applications: a New Approach via Optimal Transport.” Journal of Convex Analysis 28 (2021), No. 4, 1033–1052. https://doi.org/10.68381/jca28060

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.