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On surfaces of minimum area

Published online by Cambridge University Press:  24 October 2008

A. S. Besicovitch
A. S. Besicovitch
Affiliation:
Trinity CollegeCambridge
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Extract

This paper is devoted to the classical problem of surfaces of minimum area subtending a given closed contour.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 3 , July 1948 , pp. 313 - 334
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

† ‘On the definition and value of the area of a surface’, Quart. J. Math. 16 (1945), 86–102.Google Scholar

‡ To the usual definition of the Carathéodory two-dimensional measure it is natural to attach the constant factor ¼π so that

where E = ∑E i, and dE i < δ for all i. Throughout, d will always be used for the diameter of a set and D with a specifying suffix for the distance. Whenever the diameter of a set is mentioned it always means the Euclidean diameter, that is, the upper bound of the Euc.-distance between two points of the set. (The affix ‘Euc.-’ is used in an obvious sense throughout the paper.)

† The symbols s(M, r), S(M, r), are used respectively for the surface of the sphere with centre M and radius r, for the interior of the same sphere (the open sphere) and for the interior together with the surface of the same sphere (the closed sphere), so that .

† See the definition of identical parametric sets on p. 315.

† Besicovitch, A. S., ‘On the definition and value of the area of a surface’, Quart. J. Math. 16 (1945), 92.Google Scholar

‡ Besicovitch, A. S., ‘A general form of the covering principle’, Proc. Cambridge Phil. Soc. 42 (1946), 2.CrossRefGoogle Scholar

† Whyburn, , Analytical topology, p. 119 (5, 2).Google Scholar