# On the convex infimum convolution inequality with optimal cost function

## TL;DR

This paper proves that symmetric random variables with log-concave tails satisfy an optimal convex infimum convolution inequality, leading to nearly optimal comparison of weak and strong moments for certain random vectors.

## Contribution

It establishes the convex infimum convolution inequality with an optimal cost function for symmetric log-concave tail variables, advancing moment comparison theory.

## Key findings

- Symmetric variables with log-concave tails satisfy the inequality.
- Nearly optimal comparison of weak and strong moments achieved.
- Results apply to symmetric random vectors with independent coordinates.

## Abstract

We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails.