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 a convex function $\Phi$ : R^n $\rightarrow$ (0, +$\infty$) and the condition for the surface to be an affine hemisphere involves the 2-moment measure of $\Phi$ (a particular case of q-moment measures, i.e measures of the form ($\nabla$$\Phi$) \# ($\Phi$^{--(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 studied using the Borell-Brascamp-Lieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function $\Phi$ 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 and the optimizer turns out to be of the form $\rho$ = $\Phi$^{--(n+q)}.