Functional Versions of Lp-Affine Surface Area and Entropy Inequalities

Abstract : In contemporary convex geometry, the rapidly developing Lp-Brunn-Minkowski theory is a modern analogue of the classical Brunn-Minkowski theory. A central notion of this theory is the Lp-affine surface area of convex bodies. Here, we introduce a functional analogue of this concept, for log-concave and s-concave functions. We show that the new analytic notion is a generalization of the original Lp-affine surface area. We prove duality relations and affine isoperimetric inequalities for log-concave and s-concave functions. This leads to a new inverse log-Sobolev inequality for s-concave densities.
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Submitted on : Tuesday, January 26, 2016 - 10:29:04 PM
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Umut Caglar, Matthieu Fradelizi, Olivier Guédon, Joseph Lehec, Carsten Schütt, et al.. Functional Versions of Lp-Affine Surface Area and Entropy Inequalities. International Mathematics Research Notices, Oxford University Press (OUP), 2016, pp.1223-1250. ⟨10.1093/imrn/rnv151⟩. ⟨hal-01262626⟩

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