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Some Simple Geometrical Extremal Problems

Published online by Cambridge University Press:  03 November 2016

S. J. Taylor
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Extract

Suppose C is a convex curve bounding a plane domain of area A. l will denote the length of C and d its diameter ; that is, the upper bound of the distance between any pair of its points. It is clear that, for all convex curves C, 2d≤l≤πd. In this paper I consider classes of convex curves C with a given value of l satisfying one or more further conditions. The aim will be in each case to find the curves which enclose minimum or maximum area. The existence of a curve of a given class ∑ for which the area enclosed attains the maximum or minimum possible follows from compactness in the class of convex curves in a bounded part of the plane.

Type
Research Article
Information
The Mathematical Gazette , Volume 37 , Issue 321 , September 1953 , pp. 188 - 198
Copyright
Copyright © The Mathematical Association 1953

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References

Bull. Amer. Math. Soc., XLV (1939), 588–591