A note on Santaló inequality for the polarity transform and its reverse

Abstract:

We prove a Santaló and a reverse Santaló inequality for the class consisting of even log-concave functions attaining their maximal value $1$ at the origin, also called even geometric log-concave functions. We prove that there exist universal numerical constants $c,C>0$ such that for any even geometric log-concave function $f=e^{-\varphi }$, \[ c^{n}\cdot \left |B_{2}^{n}\right |^{2}\le \int _{\mathbb {R}^{n}}e^{-\varphi }\int _{\mathbb {R}^{n}}e^{-\varphi ^{\circ }} \le \left (\left |B_{2}^{n}\right |n!\right )^{2}\left (1+\frac {C}{n}\right ) \] where $B_{2}^{n}$ is the Euclidean unit ball of $\mathbb {R}^{n}$ and $\varphi ^{\circ }$ is the polar function of $\varphi$ (not the Legendre transform!), a transform which was recently rediscovered by Artstein-Avidan and Milman and is defined below. The bounds are sharp up to the optimal constants $c,C$.