q-Moment Measures and Applications: a New Approach via Optimal Transport

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 $\varrho_{\rm {opt}} = \varphi^{-(n + q)}$.