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

and the condition for the surface to be an affine hemisphere involves the 2-moment measure of

\varphi φ

(a particular case of

q q

-moment measures, i.e measures of the form

{(\nabla \varphi)_\# }{\varphi^{-({n + q})}} ( ∇ φ ) # φ − ( n + q )

for

q > 0 q > 0

). In Klartag's paper,

q 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 φ

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 ϱ

and the optimizer

\varrho_{\rm {opt}} ϱ o p t

turns out to be of the form

\varrho_{\rm {opt}} = \varphi^{-(n + q)} ϱ o p t = φ − ( n + q )

.