Geometric Representation of Classes of Concave Functions and Duality

Using a natural representation of a 1/ s -concave function on Rᵈ R d as a convex set in Rᵈ⁺¹, 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.