Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities

S. G. Bobkov, A. Colesanti, I. Fragalà

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa-Leindler and Brascamp-Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy-Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.

Original languageEnglish (US)
Pages (from-to)131-169
Number of pages39
JournalManuscripta Mathematica
Volume143
Issue number1-2
DOIs
StatePublished - Jan 1 2014

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