Quermassintegrals of quasi-concave functions and generalized Pr\'ekopa-Leindler inequalities

TL;DR

This paper extends the concept of quermassintegrals to quasi-concave functions, establishing properties, inequalities, and integral-geometric formulas that generalize classical convex geometric results to a functional setting.

Contribution

It introduces a functional framework for quermassintegrals, proving Steiner-type formulas and generalized Prékopa-Leindler inequalities for quasi-concave functions.

Findings

Proved a Steiner-type formula for functional quermassintegrals.

Established concavity inequalities generalizing Prékopa-Leindler and Brascamp-Lieb.

Derived integral-geometric and isoperimetric inequalities in the functional setting.

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 $\alpha$-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 Pr\'ekopa-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/Uryshon-like inequalities.