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SUMMARY:Convexity\, Optimal mass transport\, and subdifferential
DTSTART:20250610T075000Z
DTEND:20250610T085000Z
DTSTAMP:20260423T143000Z
UID:indico-event-14422@indico.math.cnrs.fr
DESCRIPTION:Speakers: Jérôme Bertrand (IMT)\n\nGiven a  compact convex 
 subset (with non-empty interior) K of Euclidean space\, the Steiner formul
 a states that the volume of the r-enlarged set K_r (namely\, the set of po
 ints at distance at most r from K) is a polynomial of r.\nThe Steiner form
 ula 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 measur
 es on the unit sphere like  the Minkowski area measure or the Aleksandrov
  curvature measure. The knowledge of one of these two measures fully chara
 cterizes K up to a simple geometric transformation (translation or dilatio
 n with respect to O). \nIn 2016\, Huang et al showed the existence of dua
 l 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 t
 o study (i.e. the support of such a measure is not the whole sphere in gen
 eral).\nIn this talk\, I will explain  how the theory of optimal mass tra
 nsport can be used to solve the above problem (existence of solutions)\, t
 hen\, if time permits\, I will discuss how tools from (convex) analysis ca
 n be used to  characterize K by such a measure (uniquess).\nThe existence
  part is available on Arxiv: "On the Gauss image problem" while the unique
 ness part is work in progress.\n\nhttps://indico.math.cnrs.fr/event/14422/
LOCATION:Salle F. Pellos (1R2-207)
URL:https://indico.math.cnrs.fr/event/14422/
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