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On Eggleston's theorem about affine diameters

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Horst Martini ,
Man Hung Nguen  and
Valeriu P. Soltan
Horst Martini
Affiliation:
Sektion Mathematik, PH Dresden, DDR.
Man Hung Nguen
Affiliation:
Mathematical Institute, Academy of Sciences BSSR, Kishinev, USSR.
Valeriu P. Soltan
Affiliation:
Mathematical Institute, Academy of Sciences BSSR, Kishinev, USSR.
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Extract

Let M be a convex body, i.e., a compact, convex set with non-empty interior, in n-dimensional Euclidean space En. A chord [a, b] of M is said to be an affine diameter of M, if, and only if, there exists a pair of (different) parallel supporting hyperplanes of that body, each containing one of the points a, b. The following result of Eggleston (cf. [1] and [2]) is well-known. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to exactly three affine diameters. In [3] this result is sharpened. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to at least two, but a finite number of affine diameters. A natural problem for the n-dimensional case, based on Eggleston's result, is the following (cf. also [4]). Is it true that the n-dimensional simplex is the only convex body in En such that through each interior point pass precisely 2n − l affine diameters? For the case of convex polytopes, i.e., convex bodies with a finite number of extreme points, we shall give a positive answer to this question.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 37 , Issue 1 , June 1990 , pp. 81 - 84
Copyright
Copyright © University College London 1990

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References

1.Eggleston, H. G.. Some properties of triangles as extremal convex curves. J. London Math. Soc, 28 (1953), 32–36.CrossRefGoogle Scholar
2.Eggleston, H. G.. Problems in Euclidean Space: Applications of Convexity (London, Pergamon Press, 1957).Google Scholar
3.Soltan, V. P. and Nguen, Man Hung. Some notes about affine diameters of convex figures. Combinatorica. To appear.Google Scholar
4.Soltan, V. P. and Nguen, Man Hung. Affine diameters of convex bodies. Geometrie und Anwendungen, Materialien zur 7. Tagung der Fachsektion Geometrie, Mathematische Gesellschaft der DDR (Berlin, 1988), 109–110.Google Scholar
5.Hammer, P. C.. Diameters of convex bodies. Proc. Amer. Math. Soc, 5 (1954), 304–306.CrossRefGoogle Scholar