Weak and strong moments of l_r-norms of log-concave vectors

TL;DR

This paper extends known results about Euclidean norms to general l_r-norms of log-concave vectors, showing their p-th moments are comparable to a sum involving weak moments, with bounds depending on r.

Contribution

It generalizes Paouris' Euclidean norm results to all l_r-norms for log-concave vectors, providing new moment comparisons.

Findings

p-th moments of l_r-norms are comparable to sums of first and weak p-th moments

Results hold for all p, r ≥ 1 with bounds proportional to r

Extends Euclidean norm results to general l_r-norms

Abstract

We show that for $p\geq 1$ and $r\geq 1$ the $p$-th moment of the $l_r$-norm of a log-concave random vector is comparable to the sum of the first moment and the weak $p$-th moment up to a constant proportional to $r$. This extends the previous result of Paouris concerning Euclidean norms.