Geometric representation of classes of concave functions and duality

TL;DR

This paper introduces a geometric framework for $s$-concave functions, deriving formulas for their polars, and extends classical inequalities like Santaló's to broader classes of functions, revealing new convexity properties.

Contribution

It provides a novel geometric representation of $s$-concave functions as convex sets and generalizes the Santaló inequality beyond the origin as the Santaló point.

Findings

Derived a formula for the integral of the $s$-polar of a $1/s$-concave function.

Proved the log-concavity of the reciprocal of the integral of the polar function.

Generalized the Santaló inequality for $s$-concave and log-concave functions.

Abstract

Using a natural representation of a $1/s$-concave function on $\mathbb{R}^d$ as a convex set in $\mathbb{R}^{d+1},$ we derive a simple formula for the integral of its $s$-polar. This leads to convexity properties of the integral of the $s$-polar function with respect to the center of polarity. In particular, we prove that that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santal\'o regions for $s$-concave and log-concave functions and generalize the Santal\'o inequality for them in the case the origin is not the Santal\'o point.