Optimal Vector Balancing for Zonotopes

A zonotope is a linear image of the cube $[–1,1]m$ for some $minmathbbN$. We show that there is a universal constant $C$ such that, for every zonotope $Zsubsetmathbb{R}^d$ and vectors $v_1,dots,v_{n}inZ$, there are signs $x_1,dots,x_{n}in–1,1$ with [ sum_{i=1}^n x_i v_i in Csqrt d, Z. ] This resolves a 2002 question of Schechtman and generalizes Spencer’s six standard deviations theorem, which corresponds to the case $Z=[–1,1]d$.