Abstract:A zonotope is a linear image of the cube $[-1,1]^m$ for some $m \in \mathbb{N}$. We show that there is a universal constant $C$ such that, for every zonotope $Z\subset \mathbb{R}^d$ and vectors $v_1,\dots,v_n\in Z$, there are signs $x_1,\dots,x_n\in\{-1,1\}$ with \[
\sum_{i=1}^n x_i v_i \in C\sqrt d\, Z. \] This resolves a 2002 question of Schechtman and generalizes Spencer's six standard deviations theorem, which corresponds to the case $Z=[-1,1]^d$.
From: Victor Reis [view email]
[v1]
Fri, 22 May 2026 17:25:00 UTC (18 KB)
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