Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T11:49:31.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of ellipsoids and polarity in convex sets

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Luis Montejano  and
Efren Morales
Luis Montejano
Affiliation:
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico, D.F.C.P. 04510.
Efren Morales
Affiliation:
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico, D.F.C.P. 04510.
Get access

Abstract

By introducing the concept of polarity in convex sets, it is possible, in a natural way, to generalize several classic characterizations of ellipsoids, showing that all of them depend upon and are related to the concept of projective centre of symmetry. Using these ideas, it is also possible to develop new characterizations of ellipsoids and to propose new problems.

MSC classification

Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 63 - 72
Copyright
Copyright © University College London 2003

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

1.Aitchison, P. W.. Petty, C. M. and Rogers, C. A.. A convex body with a false centre is an ellipsoid. Mathematika, 18 (1971). 50–59.Google Scholar
2.Arocha, J., Montejano, L. and Morales, E.. A quick proof of Hobinger-Burton-Larman's Theorem. Geom Dedicata, 63 (1996), 331–335.CrossRefGoogle Scholar
3.Bianchi, G. and Gruber, P.. Characterizations of ellipsoid. Arch. Math. 49 (1987), 334–350.CrossRefGoogle Scholar
4.Bonnesen, T. and Fenchel, W.. Theorie der Konvexen Körper (Springer, Berlin, 1934).Google Scholar
5.Busemann, H.The Geometry of Geodesies (Academic Press, New York, 1955).Google Scholar
6.Burton, G. R.Sections of convex bodies, J. London Math. Soc., 12(2) (1976), 331–336.CrossRefGoogle Scholar
7.Burton, G. R.Some characterizations of the ellipsoid Israel J. Math., 28, (1977), 339–348.Google Scholar
8.Burton, G. R. and Larman, D. G.. On a problem of Hobinger. Geom. Dedicata, 5 (1976), 31–42.Google Scholar
9.Burton, G. R. and Mani, P. A., Characterization of the ellipsoid in terms of concurrent sections. Comment. Math. Heir.. 53 (1978), 485–507.Google Scholar
10.Goodey, P. R.. Homothetic ellipsoids. Math. Proc. Camb. Phil. Soc., 93 (1983). 25–34.CrossRefGoogle Scholar
11.Goodey, P. R. and Woodcock, M. M.. The intersections of convex bodies with their translates. The Geometric Vein ed. Davis, C., Grūnbaum, B. and Sherk, F. A. (Springer Verlag, 1982).Google Scholar
12.Gruber, P. M.. Only ellipsoids have caustics. Math. Annalen, 303 (1995), 185–194.CrossRefGoogle Scholar
13.Kubota, T.. Einfache Beweise eines Satzes Uber die konvexe geschlossene Flache. Science Reports of the Töhoku Imperial University, 1st Series, 3 (1914), 235–255.Google Scholar
14.Larman, D. G.. A note on the false centre problem Mathematika, 21 (1974). 216–227.CrossRefGoogle Scholar
15.Meyer, M. and Reisner, S.. Characterization of ellipsoids by section-centroid localization. Geom. Dedicata, 31 (1989), 345–355.Google Scholar
16.Petty, C. M.. Ellipsoids, Convexity and its Applications, (eds. Gruber, P. M. and Wills, J. M.). (Birkhauser. Basel, 1985), pp. 264–276.Google Scholar