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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On congruence indices for simple closed curves
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by S. G. Wayment PDF
Proc. Amer. Math. Soc. 28 (1971), 199-207 Request permission

Abstract:

L. M. Blumenthal has defined the concept of congruence indices for point sets in his book Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. Blumenthal shows that the circle has congruence indices (3, 1) and asks if this characterizes the circle among the class of simple closed curves. In this paper it is established that various classes of simple closed curves do not have congruence indices $(3,n)$ for any $n$. Included in these classes are the polygons, simple closed curves with convex interiors and a straight line segment contained in the curve, and simple closed curves with continuous nonconstant radius of curvature on some arc. Thus any noncircular simple closed curve with congruence indices (3, 1) must be very pathological. It is shown that if a simple closed curve has positive planar Lebesgue measure, then it fails to have congruence indices $(3,n)$ for any $n$.
References
  • Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
  • L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, p. 139. MR 28 #524.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 199-207
  • MSC: Primary 52.50
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0275295-X
  • MathSciNet review: 0275295