On the distribution of distances in homogeneous compact metric spaces

2014/07/21 by Mark Herman, Herman, Mark, Jonathan Pakianathan +1
Mathematics · #05C12 #51F99 #54E45 #54E70 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG) #math.CO #math.GN #math.MG #msc:05C12 #msc:51F99 #msc:54E45 #msc:54E70

paper · pdf · doi:10.48550/arxiv.1407.5607

8 pages

arxiv created 2014/07/21 · arxiv updated 2014/07/22

Abstract

We provide a simple proof that in any homogeneous, compact metric space of diameter D, if one finds the average distance A achieved in X with respect to some isometry invariant Borel probability measure, then (D)/(2) ≤ A ≤ D. This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds. We then classify the cases where one of the extremes occurs. In particular any homogeneous compact metric space where A=(D)/(2) possesses a strict antipodal property which implies in particular that the distribution of distances in X is symmetric about (D)/(2) which is hence both mean and median of the distribution. In particular, we show that the only closed, connected, positive-dimensional Riemannian manifolds with this strict antipodal property are spheres.

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