The symmetric property tau for the Gaussian measure

TL;DR

This paper proves the symmetric property tau for the Gaussian measure using the Poincaré inequality, linking it to a functional form of the Blaschke-Santaló inequality, and clarifies its theoretical significance.

Contribution

It provides a new proof of the symmetric property tau for Gaussian measures based on the Poincaré inequality, establishing its equivalence to a functional form of the Blaschke-Santaló inequality.

Findings

Proof of the symmetric property tau using Poincaré inequality

Equivalence between tau property and Blaschke-Santaló inequality form

Theoretical insight into Gaussian measure symmetries

Abstract

We give a proof based on the Poincar\'e inequality of the symmetric property tau for the Gaussian measure. This property turns out to be equivalent to a certain functional form of the Blaschke-Santal\'o inequality, as explained in a paper by Artstein, Klartag and Milman.