Grigory Ivanov: Geometric representation of 1 / s -concave functions and duality

We will discuss a simple way of representing $1/s$-concave functions on $R^d$ as convex bodies in $R^{d+1},$ which allows us to use both standard ideas of convexity and tricks from the study of log-concave functions. We will discuss the properties of polar'' functions as an application. 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.