@TECHREPORT{Swanepoel97cardinalitiesof, author = {K. J. Swanepoel}, title = {Cardinalities Of K-Distance Sets In Minkowski Spaces}, institution = {}, year = {1997} }

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Abstract

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) points, with equality i the unit ball is a parallelotope. We solve this conjecture in the armative for all 2-dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we nd various weaker upper bounds for k-distance sets.