Selectively Balancing Unit Vectors

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 $\frac{1}{2}n\log n$.

5 pages, 2 figures. Accepted version, with significantly improved result

References in corpus (1)

• Balancing unit vectors

Cited by in corpus (1)

• Large signed subset sums