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A convex body with a false centre is an ellipsoid

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

P. W. Aitchison ,
C. M. Petty  and
C. A. Rogers
P. W. Aitchison
Affiliation:
The University of Manitoba, Winnipeg, Canada.
C. M. Petty
Affiliation:
The University of Missouri, Columbia, Missouri, U.S.A.
C. A. Rogers
Affiliation:
University College, London, U.K.
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Extract

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with a non-empty interior, then a point p of the interior of K is called a pseudo centre of K provided each two-dimensional flat through p intersects K in a section centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p of K is called a false centre if K is not centrally symmetric about p. Rogers [5] showed that a convex body (compact convex set with interior points) with a pseudo centre necessarily has a true centre of symmetry. But, as each interior point of an ellipsoid is a pseudo centre, the true centre need not necessarily coincide with the pseudo centre. Rogers conjectured that, for n ≥ 3, a convex body K with a false centre is necessarily an ellipsoid. In this paper we prove this conjecture.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 50 - 59
Copyright
Copyright © University College London 1971

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References

1.Bonnesen, T. and Fenchel, W., Theorie der konvex Körper, Ergebnisse der Mathematik (Springer Berlin, 1934 and Chelsea, New York, 1948).Google Scholar
2.Busemann, H., The geometry of geodesics (New York, 1955).Google Scholar
3.Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
4.Kubota, T., “Einfache Beweise eines Satzes Über die konvexe, geschlossene Flache”, Science Reports of the Tôhoku Imperial University, 1st Series, 3 (1914), 235–255.Google Scholar
5.Rogers, C. A., “Sections and projections of convex bodies”, Portugaliae Mathematica, 24 (1965), 99–103.Google Scholar