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The extreme points of subsets of s-concave probabilities and a geometric localization theorem

Abstract : We prove that the extreme points of the set of s-concave probability measures satisfying a linear constraint are some Dirac measures and some s-affine probabilities supported by a segment. From this we deduce that the constrained maximization of a convex functional on the s-concave probability measures is reduced to this small set of extreme points. This gives a new approach to a localization theorem due to Kannan, Lovasz and Simonovits which happens to be very useful in geometry to obtain inequalities for integrals like concentration and isoperimetric inequalities. Roughly speaking, the study of such inequalities is reduced to these extreme points.
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Submitted on : Thursday, May 3, 2012 - 11:40:50 AM
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Matthieu Fradelizi, O Guedon. The extreme points of subsets of s-concave probabilities and a geometric localization theorem. Discrete and Computational Geometry, Springer Verlag, 2004, 31 (2), pp.327--335. ⟨10.1007/s00454-003-2868-y⟩. ⟨hal-00693986⟩

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