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arXiv · 2609.31389

Unbalancing unit vectors

Zilin Jiang·Jeck Lim·Skand Parvatikar
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Abstract

We show that for every $n$ unit vectors $v_1, \dots, v_n$ in the $d$-dimensional Euclidean space, there exist signs $\varepsilon_1, \dots, \varepsilon_n \in \{\pm 1\}$ such that $\lVert \varepsilon_1 v_1 + \dots + \varepsilon_n v_n \rVert \ge \sqrt{2n - d}$, and we characterize the equality cases.

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Zilin Jiang, Jeck Lim, Skand Parvatikar. 2026-09-25. Unbalancing unit vectors. https://arxiv.org/abs/2609.31389

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