Norms of weighted sums of log-concave random vectors

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.