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Average distances in compact connected spaces

Published online by Cambridge University Press:  17 April 2009

David Yost
David Yost
Affiliation:
Department of Pure Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia.
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Abstract

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We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X with

We examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 26 , Issue 3 , December 1982 , pp. 331 - 342
Copyright
Copyright © Australian Mathematical Society 1982

References

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