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

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.