Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-27T05:14:01.705Z Has data issue: false hasContentIssue false
  • English
  • Français

A maximal Gross-Stadje number in the Euclidean plane

Published online by Cambridge University Press:  17 April 2009

F. Pillichshammer
F. Pillichshammer
Affiliation:
Institut für Mathematik, Universität Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a compact, connected Hausdorff space and f a real valued, symmetric, continuous function on X × X. Then the Gross-Stadje number r (X, f) is the unique real number with the property that for each positive integer n and for all (not necessarily distinct) x1,…,xn in X, there exists some x in X such that . This paper solves the following open question in distance geometry: What is the least upper bound g2(R2) of r (X, d2), where X ranges over all compact, connected subsets of the Euclidean plane with diameter one and where d2 denotes the squared, Euclidean distance. We show: .

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 109 - 119
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Cleary, J., Morris, S.A. and Yost, D., ‘Numerical geometry - numbers for shapes’, Amer. Math. Monthly 93 (1986), 117–127.CrossRefGoogle Scholar
[2]Gross, O., ‘The Rendezvous value of a metric space’, in Advances in Game Theory, Ann. of Math. Studies 52 (Princeton University Press, Princeton, N.J., 1964), pp. 49–53.Google Scholar
[3]Stadje, W., ‘A property of compact connected spaces’, Arch. Math. (Basel) 36 (1981), 275–280.Google Scholar
[4]Wilson, D.J., ‘A game with squared distance as payoff’, Melbourne University Mathematics Research Report 1982, no. 16.Google Scholar