# A characterization of a class of convex log-Sobolev inequalities on the real line

## TL;DR

This paper characterizes when probability measures on the real line satisfy convex log-Sobolev inequalities, linking it to transport maps and deriving dimension-free concentration bounds for convex functions.

## Contribution

It provides a necessary and sufficient condition for convex log-Sobolev inequalities on the real line using transport maps, advancing the understanding of measure concentration.

## Key findings

- Characterization of convex log-Sobolev inequalities via transport maps
- Dimension-free concentration bounds for convex functions
- Application of weak transport costs in the proof

## Abstract

We give a sufficient and necessary condition for a probability measure $\mu$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $\mu$. The main tool in the proof is the theory of weak transport costs. As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.