Asymmetric Blaschke-Santal\'o functional inequalities

TL;DR

This paper develops asymmetric functional versions of the Blaschke-Santaló inequality, connecting geometric, probabilistic, and functional analysis perspectives, and introduces an $L_p$ analogue of the center of mass.

Contribution

It introduces novel asymmetric functional inequalities related to the Blaschke-Santaló inequality and explores their geometric and probabilistic implications.

Findings

Recovered geometric inequalities with equality cases

Established a probabilistic inequality by Lutwak, Yang, and Zhang

Proposed an $L_p$ functional analogue to the center of mass

Abstract

In this work we establish functional asymmetric versions of the celebrated Blaschke-Santal\'o inequality. As consequences of these inequalities we recover their geometric counterparts with equality cases, as well as, another inequality with strong probabilistic flavour that was firstly obtained by Lutwak, Yang and Zhang. We present a brief study on an $L_p$ functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest.