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A refinement of theorems of Kirchberger and Carathéodory

Published online by Cambridge University Press:  09 April 2009

Donald Watson
Donald Watson
Affiliation:
Department of Mathematics RAAF Academy Point Cook Victoria, 3029, Australia
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I can indicate the type of refinement mentioned in the title by referring to Kirchberger's theorem [4]. Its picturesque form in the plane is: if sheep and goats are grazing in a field and for every four animals there exists a line separating the sheep from the goats then there exists such a line for all the animals. The refinement is that the words ‘every four animals’ may be replaced by ‘every four animals including an arbitraily chosen animal’; this reduces the ‘Kirchberger number’ from four to, effectively, three.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 2 , March 1973 , pp. 190 - 192
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Baker, M. J. C., Families of interesting convex sets (Ph. D. Thesis, University of Melbourne, 1968.)Google Scholar
[2]Carathéodory, C., ‘Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen’, Math. Ann. 64 (1907), 95–115.CrossRefGoogle Scholar
[3]Helly, E., ‘Über Mengen konvexen Körper mit gemeinschaftlichen Punkten’, Jber. Deutsch. Math. Verein. 32 (1923), 175–176.Google Scholar
[4]Kirchberger, P., ‘Über Tschebyschefsche Annäerungsmethoden’, Math. Ann. 57 (1903), 509–540.CrossRefGoogle Scholar