# Hadamard products and moments of random vectors

## TL;DR

This paper introduces new inequalities relating weak and strong moments of random vector norms, with applications to concentration phenomena and operator norm bounds, advancing understanding in high-dimensional probability.

## Contribution

It provides novel comparison inequalities between moments of random vectors with near-optimal constants, extending tools for analyzing concentration and operator norms.

## Key findings

- Derived comparison inequalities with universal constants.
- Applied inequalities to concentration of log-concave vectors.
- Provided bounds on p-summing norms of finite rank operators.

## Abstract

We derive new comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We discuss applications to the concentration of log-concave random vectors and bounds on $p$-summing norms of finite rank operators.