Functional versions of Lp-affine surface area and entropy inequalities

Umut Caglar, Matthieu Fradelizi, Olivier Guédon, Joseph Lehec, Carsten Schütt, Elisabeth M. Werner

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29 Scopus citations


In contemporary convex geometry, the rapidly developing Lp-Brunn-Minkowski theory is a modern analog 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 analog 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.

Original languageEnglish (US)
Pages (from-to)1223-1250
Number of pages28
JournalInternational Mathematics Research Notices
Issue number4
StatePublished - 2016
Externally publishedYes

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© The Author(s) 2015.


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