Functional versions of Lp-affine surface area and entropy inequalities

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

Caglar, Umut; Fradelizi, Matthieu; Guédon, Olivier; Lehec, Joseph; Schütt, Carsten; Werner, Elisabeth (2016), Functional versions of Lp-affine surface area and entropy inequalities, International Mathematics Research Notices, 2016, 4, p. 1223-1250. 10.1093/imrn/rnv151

Type

Article accepté pour publication ou publié

Date

2016

Journal name

International Mathematics Research Notices

Volume

2016

Number

Publisher

Duke University Press

Published in

Paris

Pages

1223-1250

Abstract (EN)

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowskitheory is a modern analogue of the classical Brunn Minkowski theory. A cornerstoneof this theory is the Lp-affine surface area for convex bodies. Here, we introducea functional form of this concept, for log concave and s-concave functions. Weshow that the new functional form is a generalization of the original Lp-affinesurface area. We prove duality relations and affine isoperimetric inequalities for logconcave and s-concave functions. This leads to a new inverse log-Sobolevinequality for s-concave densities

Subjects / Keywords

affine isoperimetric inequalities; entropy; log- Sobolev inequalities