Minimum dimension embedding of finite metric spaces
Journal of Combinatorial Theory Series A
Periods of nonexpansive operators on finite l1-spaces
European Journal of Combinatorics
Alternative proof of Sine's theorem on the size of a regular polygon in Rn with the ℓ∞ -metric
Discrete & Computational Geometry
Geometry of Cuts and Metrics
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There is a long-standing conjecture of Nussbaum which asserts that every finite set in R^n on which a cyclic group of sup-norm isometries acts transitively contains at most 2^n points. The existing evidence supporting Nussbaum's conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum's conjecture might hold more generally for abelian groups of sup-norm isometries. This paper provides evidence supporting this stronger conjecture. Among other results, we show that if @C is an abelian group of sup-norm isometries that acts transitively on a finite set X in R^n and @C contains no anticlockwise additive chains, then X has at most 2^n points.