A note on Santal\'{o} inequality for the polarity transform and its reverse

Abstract

We prove a Santal\'{o} and a reverse Santal\'{o} inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1 at the origin, also called geometric log-cancave functions. The bounds are sharp up to the optimal universal constants.Comment: 10 pages, 1 figure. Section 5 from the previous version was deleted. A characterization of an equality case in Proposition 2 was adde