On Eggleston’s theorem about affine diameters. (English) Zbl 0686.52003
It is shown that a convex d-polytope P (d\(\geq 2)\) is a simplex if and only if each interior point of P belongs to precisely \(2^ d-1\) affine diameters of P. (An affine diameter of P is a chord whose endpoints lie in different parallel supporting hyperplanes of that polytope). This generalizes, at least for convex bodies with a finite number of extreme points, a planar results of H. G. Eggleston [Problems in Euclidean space. Applications of convexity. (1957; Zbl 0083.381)].
Reviewer: H.Martini
MSC:
52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
52Bxx | Polytopes and polyhedra |
Citations:
Zbl 0083.381References:
[1] | DOI: 10.2307/2032240 · Zbl 0057.38603 · doi:10.2307/2032240 |
[2] | DOI: 10.1112/jlms/s1-28.1.32 · Zbl 0050.16602 · doi:10.1112/jlms/s1-28.1.32 |
[3] | Eggleston, Problems in Euclidean Space: Applications of Convexity (1957) |
[4] | Soltan, Geometrie und Anwendungen, Materialien zur 7. Tagung der Fachsektion Geometrie, Mathematische Gesellschaft der DDR pp 109– (1988) |
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