# Norms of weighted sums of log-concave random vectors

## TL;DR

This paper establishes upper bounds for weighted sums of log-concave random vectors, providing an alternative proof for known lower bounds and exploring applications in randomized vector balancing.

## Contribution

It introduces new upper bounds for multi-integral norms of weighted sums of log-concave vectors and offers an alternative proof for existing lower bounds, with applications to vector balancing.

## Key findings

- Derived upper bounds for the multi-integral expression in isotropic convex bodies.
- Provided an alternative proof for the sharp lower bound by Gluskin and Milman.
- Applied results to randomized vector balancing problems.

## Abstract

Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to "randomized" vector balancing problems.