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Paths in the one-skeleton of a convex body

Published online by Cambridge University Press:  26 February 2010

D. G. Larman  and
C. A. Rogers
D. G. Larman
Affiliation:
University College, London, W.C.1.
C. A. Rogers
Affiliation:
University College, London, W.C.1.
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A general convex body in Euclidean space can be approximated by smooth convex bodies, and many results, arising first in the differential geometry of smooth convex bodies, have been extended to yield corresponding results for general convex bodies. Although convex bodies can be approximated by convex polyhedra, very little of the rich theory of convex polyhedra has been extended to general convex bodies. In this paper, we extend the concept of the one-skeleton of a convex polytope to yield the concept of the one-skeleton of a general convex body. We investigate the connectivity properties of this one-skeleton, and we extend a result of Balinski [1], on paths in the one-skeleton of a convex polytope to the class of convex bodies.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 293 - 314
Copyright
Copyright © University College London 1970

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References

1.Balinski, M., “On the graph structure of convex polyhedra in n space”, Pacific J. Math., 11 (1961), 431–434.Google Scholar
2.Grünbaum, B., Convex polytopes (Wiley, New York, 1967).Google Scholar
3.Menger, K., “Zur allgemeinen Kurventheorie”, Fund. Math., 10 (1927), 96–115.Google Scholar
4.Whitney, H., “Congruent graphs and the connectivity of graphs”, Amer. J. Math., 54 (1932), 150–168.CrossRefGoogle Scholar
5.Ore, O., Theory of graphs, Amer. Math. Soc. Colloq. Publs., 38 (1962).Google Scholar
6.Rutt, N. E., “Concerning the cut points of a continuous curve when the arc-curve AB contains exactly N independent arcs”, Amer. J. Math., 51 (1929), 217–246.Google Scholar
7., G. Nöbeling “Eine vershärfung des n-Beinsatzes”, Fund. Math., 18 (1931), 23–38.Google Scholar
8.Menger, K., Kurventheorie (Teubner, Berlin-Leipzig, 1932).Google Scholar
9.Zippin, L., “Independent arcs of a continuous curve”, Ann. of Math., 34 (1933), 95–113.CrossRefGoogle Scholar
10.Whyburn, G. T., “On n-arc connectedness”, Trans. Amer. Math. Soc., 63 (1948), 452–456.Google Scholar
11.Ewald, G., Larman, D. G. and Rogers, C. A., “The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space”, Mathematika, 17 (1970), 1–20.CrossRefGoogle Scholar
12.Whyburn, G. T., Analytic topology, Amer. Math. Soc. Colloq. Publs., 28 (1962).Google Scholar
13.Straszewicz, S., “Über exponierte Punkte abgeschlossener Punktmenger”, Fund. Math., 24 (1935), 139–143.Google Scholar