Support functions and mean width for {\alpha}-concave functions

Abstract

In this paper we extend some notions, previously defined for log-concave functions, to the larger domain of so-called {\alpha}-concave functions. We begin with a detailed discussion of support functions - first for log-concave functions, and then for general {\alpha}-concave functions. We continue by defining mean width, and proving some basic results such as an Urysohn type inequality. Finally, we demonstrate how such geometric results can imply Poincar\'e type inequalities.