2012/10/23 by Sergey G. Bobkov, Bobkov, Sergey, Andrea Colesanti +3 · 4 citations
Mathematics · Medicine · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities
paper · doi:10.48550/arxiv.1210.6364
openalex publication_date 2012/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/Uryshon-like inequalities.