Functional Versions of L p -Affine Surface Area and Entropy Inequalities
Umut Caglara b✉, Matthieu Fradelizic d, Olivier Guédonc d, Joseph Lehece f, Carsten Schüttg h, Elisabeth M. Wernera b i j
- aDepartment of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
- b1Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
- cUniversité Paris Est, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-77454 Marne-la-Vallée, France
- d2Université Paris Est, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-77454 Marne-la-Vallée, France
- eUniversité Paris-Dauphine, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France
- f3Université Paris-Dauphine, CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France
- gMathematisches Institut, Universität Kiel, 24105 Kiel, Germany
- h4Mathematisches Institut, Universität Kiel, 24105 Kiel, Germany
- iUniversité de Lille 1, UFR de Mathématique, 59655 Villeneuve d'Ascq, France
- j5Université de Lille 1, UFR de Mathématique, 59655 Villeneuve d'Ascq, France
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Abstract
In contemporary convex geometry, the rapidly developing |$L_p$|-Brunn–Minkowski theory is a modern analog of the classical Brunn–Minkowski theory. A central notion of this theory is the |$L_p$|-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 |$L_p$|-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.