The Vector Balancing Constant for Zonotopes

TL;DR

This paper establishes bounds on the vector balancing constant for zonotopes, showing it is roughly proportional to the square root of the dimension, and provides constructive algorithms for these bounds.

Contribution

The paper proves new bounds on the vector balancing constant for zonotopes, extending geometric inequalities and generalizing Spencer's theorem with polynomial-time algorithms.

Findings

Bound of rac{rac{rac{d}{}}{}}{ ext{log log log d}} for rac{rac{rac{rac{vb}(K,K)}{}}{}}

Extension of Vaaler's theorem to zonotopes

Polynomial-time algorithms for constructing the colorings

Abstract

The vector balancing constant $\mathrm{vb}(K,Q)$ of two symmetric convex bodies $K,Q$ is the minimum $r \geq 0$ so that any number of vectors from $K$ can be balanced into an $r$-scaling of $Q$. A question raised by Schechtman is whether for any zonotope $K \subseteq \mathbb{R}^d$ one has $\mathrm{vb}(K,K) \lesssim \sqrt{d}$. Intuitively, this asks whether a natural geometric generalization of Spencer's Theorem (for which $K = B^d_\infty$) holds. We prove that for any zonotope $K \subseteq \mathbb{R}^d$ one has $\mathrm{vb}(K,K) \lesssim \sqrt{d} \log \log \log d$. Our main technical contribution is a tight lower bound on the Gaussian measure of any section of a normalized zonotope, generalizing Vaaler's Theorem for cubes. We also prove that for two different normalized zonotopes $K$ and $Q$ one has $\mathrm{vb}(K,Q) \lesssim \sqrt{d \log d}$. All the bounds are constructive and the corresponding colorings can be computed in polynomial time.