Sets of unit vectors with small pairwise sums

Details Sets of unit vectors with small pairwise sums Sets of unit vectors with small pairwise sums

2010-06-05

arxiv.org/abs/1006.1051v1

Quaestiones Mathematicae 23 (2000) 383-388

Mathematics

Metric Geometry

6 pages. Old paper of 10 years ago

Scientific paper

10.2989/16073600009485985

We study the sizes of delta-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most delta. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. F\"uredi, J. C. Lagarias, F. Morgan, 1991). We show that the maximum size of a delta-additive set over all normed spaces of dimension d grows exponentially in d for fixed delta>2/3, stays bounded for delta<2/3, and grows linearly at the threshold delta=2/3. Furthermore, the maximum size of a 2/3-additive set in d-dimensional normed space has the sharp upper bound of d, with the single exception of spaces isometric to three-dimensional l^1 space, where there exists a 2/3-additive set of four unit vectors.

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