*Accepted Paper*

**Inserted:** 17 jun 2020

**Last Updated:** 22 sep 2020

**Journal:** J. Conv. An.

**Year:** 2020

**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 a convex function $\varphi :{{\mathbb R}^n} \to \left( {0, + \infty } \right)$ 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 ${\left( {\nabla \varphi } \right)_\# }{\varphi ^{ - \left( {n + q} \right)}}$ 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 $\varphi$ 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 $\varrho_{\rm {opt}} = {\varphi^{ - \left( {n + q} \right)}}$.

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