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Large signed sums of unit vectors: the first linearly dependent case
Abstract
We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given d+1 unit vectors u1,…,ud+1 in ℝd, with d≥2, the problem asks for the smallest possible value of maxεi=±1 |∑i=1d+1εi ui|. We prove that this value is √(d+2). We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.
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BibTeX
@misc{pinasco2026large,
title = {{Large signed sums of unit vectors: the first linearly dependent case}},
author = {Damián Pinasco},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21101},
eprint = {2609.21101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21101},
}
RIS
TY - UNPB
TI - Large signed sums of unit vectors: the first linearly dependent case
AU - Damián Pinasco
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21101
UR - https://arxiv.org/abs/2609.21101
AB - We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given d+1 unit vectors u_1,…,u_d+1 in ℝ^d, with d≥2, the problem asks for the smallest possible value of max_ε_i=±1 |∑_i=1^d+1ε_i u_i|. We prove that this value is √(d+2). We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.
ER -