# Concentration between L\'evy's inequality and the Poincar\'e inequality for log-concave densities

## TL;DR

This paper explores strong concentration phenomena for log-concave measures, revealing results that surpass classical inequalities and relate to the hyperplane conjecture through moments of projections.

## Contribution

It establishes new concentration bounds for the expectation of absolute projections of log-concave measures, linking these to fundamental open problems like the hyperplane conjecture.

## Key findings

- Strong concentration of $ heta o ext{E}|X heta|$ for log-concave measures
- Potential implications for the hyperplane conjecture via third moments
- Surpassing classical Lévý and Poincaré inequalities in this context

## Abstract

Given a suitably normalized $X\in\mathbb{R}^n$ we observe that the function $\theta\mapsto\mathbb{E}|X\cdot\theta|$, defined for $\theta\in S^{n-1}$, admits surprisingly strong concentration far surpassing what is expected on account of L\'evy's isoperimetric inequality. Among the measures to which the above holds are all log-concave measures, for which a solution of the similar problem concerning the third marginal moments $\theta\mapsto\mathbb{E} (X\cdot \theta)^3$ would imply the hyperplane conjecture.