Geometric Representation of Classes of Concave Functions and Duality

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 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.