Optimal Vector Balancing for Zonotopes

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 \(Zsubset mathbb{R}^d\) and vectors \(v_1,dots,v_nin Z\), there are signs \(x_1,dots,x_nin{-1,1}\) 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 \(Z=[-1,1]^d\).