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

TL;DR

This paper extends geometric notions like support functions and mean width from log-concave to {\alpha ext{-}concave} functions, deriving inequalities and inequalities implications for this broader class.

Contribution

It introduces support functions and mean width for {\alpha ext{-}concave} functions, generalizing concepts from log-concave functions and establishing related inequalities.

Findings

Support functions defined for {\alpha ext{-}concave} functions.

Urysohn type inequality established.

Connections to Poincaré inequalities demonstrated.

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.