Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T19:23:12.636Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetrical sets of constant width and their partitions

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
C. A. Rogers
Affiliation:
University College, London.
Get access

Extract

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 105 - 111
Copyright
Copyright © University College London 1971

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.Borsuk, K., “Drei Satze über die n-dimensionale euklidische Sphäre”, Fund. Math., 20 (1033), 177–190.CrossRefGoogle Scholar
2.Hadwiger, H., “ÜOberdeckung einer Menge durch Mengen kleineren Durchmessers”, Comment. Math. Helv. 18 (1945/1946) 73–75.CrossRefGoogle Scholar
3.Hadwiger, H., “Mitteilung betreffend meine Note:. Übetdeckuag eiaer Menge durch Mengen kleineren Durchmessers”, Comment. Math. Helv. 19 (1946/1947) 161–165.CrossRefGoogle Scholar
4.Eggleston, H. G., “Covering a three-dimensional set with sets of smaller diameter,” J. London Math.Soc., 30 (1955), 11–24.CrossRefGoogle Scholar
5.Grünbaum, B., “A simple proof of Borsuk's conjecture in three dimensions”, Proc. Cambridge Phil. Soc., 53 (1957) 776–778.CrossRefGoogle Scholar
6.Heppes, A., “On the splitting of point sets in three space into the union of sets of smaller diameter”, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 7 (1957), 413–416. (Hungarian)Google Scholar
7.Eggleston, H. G., Convexity (Cambridge, 1958)CrossRefGoogle Scholar
8.Jung, H. W. E., “Ueber die kleinste Kugel, die eine räumliche Figur einschliesst,” J. reine u. ange Math., 123 (1901), 241–257.Google Scholar