Cardinalities of k-distance sets in Minkowski spaces

Details Cardinalities of k-distance sets in Minkowski spaces Cardinalities of k-distance sets in Minkowski spaces

2007-12-06

arxiv.org/abs/0712.0953v1

Discrete Mathematics 197/198 (1999) 759-767

Mathematics

Metric Geometry

7 pages, 2 figures

Scientific paper

10.1090/S0002-9939-96-03370-9

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we find various weaker upper bounds for k-distance sets.

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