Convexity, Optimal mass transport, and subdifferential

by Jérôme Bertrand (IMT)

Tuesday Jun 10, 2025, 9:50 AM → 10:50 AM Europe/Paris

Salle F. Pellos (1R2-207)

Given a  compact convex subset (with non-empty interior) K of Euclidean space, the Steiner formula states that the volume of the r-enlarged set K_r (namely, the set of points at distance at most r from K) is a polynomial of r.

The Steiner formula admits localized versions, for instance by fixing a point O within K and considering the intersection of K_r with  measurable cones with apex O. A consequence of this localisation principle is the existence of measures on the unit sphere like  the Minkowski area measure or the Aleksandrov curvature measure. The knowledge of one of these two measures fully characterizes K up to a simple geometric transformation (translation or dilation with respect to O). 

In 2016, Huang et al showed the existence of dual measures for such a K by using a Steiner-like formula. A fair amount of these new measures satisfy a property similar to the one satisfied by the Aleksandrov curvature measure but in a local form that makes them harder to study (i.e. the support of such a measure is not the whole sphere in general).

In this talk, I will explain  how the theory of optimal mass transport can be used to solve the above problem (existence of solutions), then, if time permits, I will discuss how tools from (convex) analysis can be used to  characterize K by such a measure (uniquess).

The existence part is available on Arxiv: "On the Gauss image problem" while the uniqueness part is work in progress.