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In 2017, Bo'az Klartag obtained a new result in differential geometry on theexistence of affine hemisphere of elliptic type. In his approach, a surface isassociated with every a convex function $\Phi$ : R^n $\rightarrow$ (0,+$\infty$) and the condition for the surface to be an affine hemisphereinvolves the 2-moment measure of $\Phi$ (a particular case of q-momentmeasures, i.e measures of the form ($\nabla$$\Phi$) \# ($\Phi$^{--(n+q)}) for q> 0). In Klartag's paper, q-moment measures are studied through a variationalmethod requiring to minimize a functional among convex functions, which isstudied using the Borell-Brascamp-Lieb inequality. In this paper, we attack thesame problem through an optimal transport approach, since the convex function$\Phi$ is a Kantorovich potential (as already done for moment measures in aprevious paper). The variational problem in this new approach becomes theminimization of a local functional and a transport cost among probabilitymeasures and the optimizer turns out to be of the form $\rho$ =$\Phi$^{--(n+q)}.

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