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VARIATION ON A THEOREM BY CARATHÉODORY

Part of: General convexity

Published online by Cambridge University Press:  10 December 2009

Leonard J. Schulman
Leonard J. Schulman*
Affiliation:
Caltech, MC305-16, Pasadena, CA 91125, U.S.A. (email: [email protected])
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Abstract

Carathéodory’s theorem on small witnesses for convex hulls of sets is shown to have a natural analogue for finitely supported measures. Contrast is drawn with the much larger witnesses required for multisets, as shown by Bárány and Perles.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 169 - 172
Copyright
Copyright © University College London 2010

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References

[1]Bárány, I. and Larman, D. G., Convex bodies, economic cap coverings, random polytopes. Mathematika 35 (1988), 274–291.CrossRefGoogle Scholar
[2]Bárány, I. and Perles, M., The Caratheodory number for the k-core. Combinatorica 10(2) (1990), 185–194.CrossRefGoogle Scholar
[3]Blaschke, W., Vorlesungen über Differentialgeometrie II, Springer (Berlin, 1923).Google Scholar
[4]Carathéodory, C., Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64 (1907), 95–115.CrossRefGoogle Scholar
[5]Carathéodory, C., Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32 (1911), 193–217.CrossRefGoogle Scholar
[6]Danzer, L., Grünbaum, B. and Klee, V., Helly’s Theorem and its Relatives (Proceedings Symposia in Pure Mathematics, Volume VII Convexity), American Mathematical Society (Providence, RI, 1963), 101–181.Google Scholar
[7]Dupin, C., Application de Géometrie et de Méchanique à la Marine, aux Ponts et Chaussées (1822).Google Scholar
[8]Leichtweiss, K., Über ein Formel Blaschkes zur Affinoberfläche. Studia Sci. Math. Hungar. 21 (1986), 453–474.Google Scholar
[9]Matoušek, J., Lectures on Discrete Geometry, Springer (Berlin, 2002).CrossRefGoogle Scholar
[10]Rado, R., A theorem on general measure. J. London Math. Soc. 21 (1946), 291–300.CrossRefGoogle Scholar
[11]Schutt, C. and Werner, E., The convex floating body. Math. Scand. 66 (1990), 275–290.CrossRefGoogle Scholar
[12]Stancu, A., The floating body problem. Bull. London Math. Soc. 38 (2006), 839–846.CrossRefGoogle Scholar