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Applications of outwardly simple line families to plane convex sets

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
Corpus Christi College, Cambridge
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Extract

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 353 - 356
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Besicovitch, A. S.On semicircles inscribed into sets of constant width. Proc. Symp. Pure. Math., no. VII, Convexity, pp. 15–18 (Amer. Math. Soc. 1963).Google Scholar
(2)GrünbaUm, B.Measures of symmetry for convex sets. Proc. Symp. Pure Math., no. VII, Convexity, pp. 233–270 (Amer. Math. Soc. 1963).CrossRefGoogle Scholar
(3)Hammer, P. C. and Smith, T. J.Conditions equivalent to central symmetry of convex curves. Proc. Cambridge Philos. Soc. 60 (1964), 779–785.CrossRefGoogle Scholar
(4)Hammer, P. C. and Sobczyk, A.Planar line families I. Proc. Amer. Math. Soc. 4 (1953), 226–233.CrossRefGoogle Scholar