Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-14T18:26:15.290Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
2Larman, D. G., A note in the false centre theorem. Mathematika 21 (1974), 216–227.CrossRefGoogle Scholar
3Montejano, L., Two applications of topology to convex geometry (preprint).Google Scholar
4Montejano, L., Convex bodies with homothetic sections. Bull. London Math. Soc. 23 (1991), 381–386.CrossRefGoogle Scholar
5Montejano, L. and Morales, E., Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika (to appear).Google Scholar
6Rogers, C. A., Sections and projections of convex bodies. Portugaliae Math. 24 (1965), 99–103.Google Scholar
7Rogers, C. A., An equichordal problem. Geom. Dedicata 10 (1981), 73–80.CrossRefGoogle Scholar