Selectively Balancing Unit Vectors

TL;DR

This paper establishes that in high-dimensional space, the smallest set of unit vectors needed to ensure a specific balancing property grows asymptotically as half of n log n, revealing fundamental geometric limits.

Contribution

It provides the first asymptotic characterization of the minimum number of vectors needed for selective balancing in Euclidean spaces.

Findings

Minimum number of vectors for selective balancing is asymptotically (1/2) n log n.

Proves a fundamental geometric property in high-dimensional vector sets.

Advances understanding of vector balancing in Euclidean spaces.

Abstract

A set $U$ of unit vectors is selectively balancing if one can find two disjoint subsets $U^+$ and $U^-$, not both empty, such that the Euclidean distance between the sum of $U^+$ and the sum of $U^-$ is smaller than $1$. We prove that the minimum number of unit vectors that guarantee a selectively balancing set in $\mathbb{R}^n$ is asymptotically $\frac{1}{2} n \log n$.