Colorful Vector Balancing

TL;DR

This paper improves classical bounds on vector balancing constants in Euclidean and maximum norms for specific cases, using linear algebra, probability, and Gaussian random walks, with results proven to be sharp.

Contribution

It extends classical vector balancing estimates for p=2 and p=∞ norms, providing sharp bounds for vector families within Euclidean and maximum norm balls.

Findings

Bounds are sharp for p=2 and p=∞ norms.

Selected vectors sum to at most O(√d) in these norms.

Proofs combine algebraic, probabilistic, and Gaussian methods.

Abstract

We extend classical estimates for the vector balancing constant of $\mathbb{R}^d$ equipped with the Euclidean and the maximum norms proved in the 1980's by showing that for $p =2$ and $p=\infty$, given vector families $V_1, \ldots, V_n \subset B_p^d$ with $0 \in \sum_{i=1}^n \mathrm{conv}\, V_i$, one may select vectors $v_i \in V_i$ with $ \| v_1 + \ldots + v_n \|_2 \leq \sqrt{d}$ for $p=2$, and $ \| v_1 + \ldots + v_n \|_\infty \leq O(\sqrt{d}) $ for $p = \infty$. These bounds are sharp and asymptotically sharp, respectively, for $n \geq d$. The proofs combine linear algebraic and probabilistic methods with a Gaussian random walk argument.