On cost-induced Santal\'o-type inequalities in Polish measure spaces

Abstract

We introduce a framework for establishing Blaschke-Santal\'o-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santal\'o-type. We call these equivalent inequalities cost-Santal\'o inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santal\'o inequalities, including on the space of rectangular matrices and a functional sine Santal\'o inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santal\'o inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santal\'o inequality, which we explicitly instantiate for matrix spaces.