Blaschke-Santal\'o inequalities for Minkowski and Asplund endomorphisms

TL;DR

This paper explores inequalities related to Minkowski and Asplund endomorphisms, establishing new isoperimetric inequalities, analyzing their invariance properties, and extending results to functional analogues for log-concave functions.

Contribution

It introduces a family of isoperimetric inequalities derived from Minkowski endomorphisms, highlighting the uniqueness of the Blaschke-Santaló inequality and extending these concepts to functional settings.

Findings

The Blaschke-Santaló inequality is the strongest among the derived inequalities.

Extension to weakly monotone Minkowski endomorphisms is impossible.

Functional inequalities for log-concave functions generalize the classical Blaschke-Santaló inequality.

Abstract

It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one - the Blaschke-Santal\'o inequality - turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes the functional Blaschke-Santal\'o inequality.