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A note on the false centre problem

Published online by Cambridge University Press:  26 February 2010

D. G. Larman
Affiliation:
Dept. of Mathematics, University College London, Gower Street, London WC1E 6BT.
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If K is a set in n-dimensional Euclidean space En, n ≥ 2, with non-empty interior, then a point p of En is called a pseudo-centre of K provided tha each two dimensional flat through p intersects K in a section, which is either empt or centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p is called a false centre of K, if K is not centrally symmetric about p. A dee result of Aitchison, Petty and Rogers [1] asserts that, if K is a convex body in E and p is a false centre of K, with p in the interior, int K, of K, then K is an ellipsoid Recently J. Höbinger [2] extended this theorem to any smooth convex body K witl a false centre p anywhere in En. At a recent meeting in Oberwolfach, he asked if the condition of smoothness can be omitted and the purpose of this note is to prove sucl a result.

Type
Research Article
Copyright
Copyright © University College London 1974

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References

1.Aitchison, P. W., Petty, C. M. and Rogers, C. A.. “A convex body with a false centre is an ellipsoid”, Mathematika, 18 (1971), 5059.CrossRefGoogle Scholar
2.Höbinger, J.. “Üeber eine Kennzeichnung von Aitchison, Petty und Rogers”, to appear.Google Scholar
3.Anderson, R. D. and Klee, V. L.. “Convex functions and upper semi-continuous functions”, Duke Math., 19 (1952), 349357.CrossRefGoogle Scholar
4.Rogers, C. A.. “Sections and projections of convex bodies”, Portugaliae Mathematica, 24 (1965), 99103.Google Scholar
5.Busemann, H.. The geometry of geodesies (New York, 1955).Google Scholar