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Abstract: We study the sizes of delta-additive sets of unit vectors in a d-dimensionalnormed space: the sum of any two vectors has norm at most delta. One-additivesets originate in finding upper bounds of vertex degrees of Steiner MinimumTrees 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 overall 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 normedspace has the sharp upper bound of d, with the single exception of spacesisometric to three-dimensional l^1 space, where there exists a 2-3-additive setof four unit vectors.



Author: Konrad J. Swanepoel

Source: https://arxiv.org/



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