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Shaken False Centre Theorem, I

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

L. Montejano  and
E. Morales-Amaya
L. Montejano
Affiliation:
Instituto de Matemáticas, UNAM, Circuito exterior, C.U., México D.F., 04510, México. E-mail: luis@math.unam.mx
E. Morales-Amaya
Affiliation:
Centro de Investigación en Matemáticas, A.C., A.P., 402, Guanajuato, Gto., C.P. 3600, México. E-mail: efren@cimat.mx
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Extract

Let K be a convex body of dimension at least 3, and let p0 be a point. If every section of K through p0 is centrally symmetric, then Rogers proved in [6] that K is centrally symmetric, although p0 may not be the centre of K. If this is the case, then Aitchison, Petty and Rogers [1] and Larman [2] proved that K must be an ellipsoid. Suppose now that, for every direction, we can choose continuously a section of K that is centrally symmetric; if K is strictly convex, then Montejano [3] proved that K must be centrally symmetric. Consider now the following example. Let D be a (euclidean) ball centred at the origin from which two symmetric caps are deleted. Then D is centrally symmetric with respect to the origin, and has a lot of circular sections whose centre is not the origin. In fact, we can choose continuously, for every direction, a section of D which is centrally symmetric, in such a way that not all these sections pass through the origin. Nevertheless, no matter how we choose these sections, there are always necessarily many of them that do pass through the origin. For those sections, of course, we have not imposed any condition, which explains the fact that D is not a quadric elsewhere.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 41 - 46
Copyright
Copyright © University College London 2007

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References

1Aitchison, P. W., Petty, C. M. and Rogers, C. A., A convex body with a false centre is an ellipsoid. Mathematika 18 (1971), 50–59.CrossRefGoogle Scholar
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