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Equichordal curves

Published online by Cambridge University Press:  14 November 2011

F.J.Craveiro de Carvalho  and
S.A. Robertson
F.J.Craveiro de Carvalho
Affiliation:
Departamento de Matemática, Universidade de Coimbra, 3000 Coimbra, Portugal
S.A. Robertson
Affiliation:
Faculty of Mathematical Studies, Southampton University, Southampton SO9 5NH, U.K.
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Synopsis

In this paper we extend the notion of equichordal curve to closed simple curves in ℝn. Although it is not known if an equichordal curve can have more than one fulcrum, we show that, for plane curves, any fulcrum is inside the curve. We establish connections with the theories of transnormal and self-parallel curves and lower bounds for the length and chordal area are obtained. Such bounds are the best possible.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 1-2 , 1988 , pp. 85 - 91
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Blaschke, W., Rothe, H. and Weitzenbück, R.. Aufgabe 552. Arch. Math. u. Phys. 27 (1917), 82.Google Scholar
2Carter, S.. A class of compressible embeddings. Proc. Cambridge Phil. Soc. 65 (1969), 23–26.CrossRefGoogle Scholar
3Chakerian, G. D.. A characterization of curves of constant width. Amer. Math. Monthly 81 (1974), 153–155.CrossRefGoogle Scholar
4Connett, J. E.. A generalisation of the Borsuk-Ulam theorem. J. London Math. Soc. 7 (1973), 64–66.CrossRefGoogle Scholar
5Carvalho, F. J. Craveiro de and Robertson, S. A.. Self-parallel curves (to appear).Google Scholar
6Fujiwara, M.. Über die mittelkurve zweier geschlossenen konvexen kurven in bezug auf einen punkt. Tôhoku Math. J. 10 (1916), 99–103.Google Scholar
7Guillemin, V. and Pollack, A.. Differential Topology (New York:Prentice-Hall, 1974).Google Scholar
8Kelly, P. J.. Curves with a kind of constant width. Amer. Math. Monthly 64 (1957), 333–336.CrossRefGoogle Scholar
9Klee, V.. Can a plane convex body have two equichordal points?. Amer. Math. Monthly 76 (1969), 54–55.CrossRefGoogle Scholar
10Klee, V.. Some unsolved problems in plane geometry. Math. Mag. 52 (1979), 131–145.CrossRefGoogle Scholar
11Millman, R. S. and Parker, G. D.. Elements of Differential Geometry (New York: Prentice-Hall, 1977).Google Scholar
12Robertson, S. A.. Smooth curves of constant width and transnormality. Bull. London Math. Soc. 16 (1984), 264–274.CrossRefGoogle Scholar
13Rutishauser, H. and Samelson, H.. Sur le rayon d'une sphère dontla surface contient une courbe fermée. C. R. Acad. Sci. 227 (1948), 755–757.Google Scholar
14Tiercy, G.. Sur les courbes orbiformes. Leur utilisation en mèchanique. Tôhoku Math. J. 18 (1920), 90–115.Google Scholar