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8 papers · 1 filter
Unit distances and diameters in Euclidean spaces
Konrad J Swanepoel
We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, fo…
A lower bound for the equilateral number of normed spaces
Konrad J Swanepoel, Rafael Villa
We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity is at most 3/2, then there exist n+1 equidistant points in X. By a well-known res…
Low-degree minimal spanning trees in normed spaces
Horst Martini, Konrad J Swanepoel
We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the s…
The local Steiner problem in finite-dimensional normed spaces
Konrad J Swanepoel
We develop a general method of proving that certain star configurations in finit e-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlo…
Upper bounds for edge-antipodal and subequilateral polytopes
Konrad J Swanepoel
A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of…
Equilateral sets in finite-dimensional normed spaces
Konrad J. Swanepoel
This is an expository paper on the largest size of equilateral sets in finite-dimensional normed spaces.