Geometric representation of classes of concave functions and duality

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 Santaló regions for $s$-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.