On the infimum convolution inequalities with improved constants

TL;DR

This paper improves the constants in infimum convolution inequalities for certain distributions, showing the exponential distribution satisfies IC with constant 2 and applying this to enhance concentration inequalities.

Contribution

It demonstrates the exponential distribution's IC constant is 2, not 1, and uses this to refine constants in IC inequalities for log-concave measures and Talagrand's concentration.

Findings

Exponential distribution satisfies IC with constant 2.

Linear functions are not extremal in Maurey's property ( au).

Improved constants in concentration inequalities for product measures.

Abstract

The goal of the article is to improve constants in the infimum convolution inequalities (IC for short) which were introduced by R. Lata{\l}a and J.O. Wojtaszczyk. We show that the exponential distribution satisfies IC with constant $2$ but not with constant $1$, which implies that linear functions are not extremal in Maurey's property $(\tau)$. Using transport of measure we use this result to better constants in the IC inequalities for product symmetric log-concave measures as well as in the Talagrand's two level concentration inequality for the exponential distribution.