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

By Sergey G. Bobkov, Andrea Colesanti, Ilaria Fragalà. 2012

62 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a8740f3f11d2d9e425dec2ca1ce10f56fc85b1e2

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Valuations on Convex Functions (2017, background): “Moreover, classical functionals for convex sets including the intrinsic volumes have been extended to the space of quasi-concave functions in [7] and [28] (see also [9, 18]).”
Rogers–Shephard inequality for log-concave functions (2014, background): “In [18,22,30], the case of the perimeter and the mean width is considered while, in [13], a different definition is given for all the quermaßintegrals.”