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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

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