The Scaled Polarity transform and related inequalities

In this paper we deal with generalizations of the Mahler volume product for log-concave functions. We show that the polarity transform $\mathcal A$ can be rescaled so that the Mahler product it induces has upper and lower bounds of the same asymptotics. We discuss a similar result for the $\mathcal J$ transform. As an application, we extend the König-Milman duality of entropy result to the class of geometric log-concave functions.