Functional Versions of Lp-Affine Surface Area and Entropy Inequalities

Umut Caglar (1) , Matthieu Fradelizi (2) , Olivier Guédon (2) , Joseph Lehec (3) , Carsten Schütt (4) , Elizabeth Werner (1)

1 Case Western Reserve University [Cleveland]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 Mathematisches Seminar [Kiel]

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.

Domaines: Analyse fonctionnelle [math.FA]

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, 2016, 4, pp.1223-1250. ⟨10.1093/imrn/rnv151⟩. ⟨hal-01262626⟩