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arXiv · 1906.03719

Norms of weighted sums of log-concave random vectors

Giorgos Chasapis·Apostolos Giannopoulos·Nikos Skarmogiannis
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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.

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Giorgos Chasapis, Apostolos Giannopoulos, Nikos Skarmogiannis. 2019-06-09. Norms of weighted sums of log-concave random vectors. https://arxiv.org/abs/1906.03719

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