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Conditions equivalent to central symmetry of convex curves

Published online by Cambridge University Press:  24 October 2008

P. C. Hammer  and
T. Jefferson Smith
P. C. Hammer
Affiliation:
University of Wisconsin, Madison and Carnegie Institution, Washington
T. Jefferson Smith
Affiliation:
University of Wisconsin, Madison and Carnegie Institution, Washington
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Extract

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:

(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)

(2) Each line halving the area of C is a diametral line.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 779 - 785
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Besicovitch, A. S.A problem on a circle. J. London Math. Soc. 36 (1961), 21–56.Google Scholar
(2)Gruenbaum, B. Measures of symmetry of convex curves, from Proceedings of symposia in pure mathematics VII. Convexity (American Math. Soc., 1963).Google Scholar
(3)Hammer, P. C.Convex bodies associated with a convex body. Proc. American Math. Soc. 2 (1951), 522–525.CrossRefGoogle Scholar
(4)Hammer, P. C.Diameters of convex bodies. Proc. American Math. Soc. 5 (1954), 304–306.CrossRefGoogle Scholar
(5)Hammer, P. C. Convex curves of constant Minkowski breadth, from Proceedings of symposia in pure mathematics VII. Convexity (American Math. Soc., 1963).Google Scholar
(6)Hammer, P. C. and Sobczyk, A. F.Planar line families. I. Proc. American Math. Soc. 4 (1953), 226–233.CrossRefGoogle Scholar
(7)Hammer, P. C. and Sobczyk, A. F.Planar line families. II. Proc. American Math. Soc. 4 (1953), 341–349.CrossRefGoogle Scholar
(8)Smith, T. J. Planar line families (Ph.D. Thesis; University of Wisconsin, 1961). Unpublished, but available as a report.Google Scholar
(9)Zindler, K.Ueber konvexe Gebilde. Monatsh. Math.-Phys. 31 (1921), 21–56.Google Scholar