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Parametric surfaces

II. Lower semi-continuity of the area

Published online by Cambridge University Press:  24 October 2008

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

In 1914 Carathéodory defined m–dimensional measure in n–dimensional space. He considered one-dimensional measure as a generalization of length and he proved that the length of a rectifiable curve coincides with its one-dimensional measure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 1 , January 1949 , pp. 14 - 23
Copyright
Copyright © Cambridge Philosophical Society 1949

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References

† ‘Über das lineare Mass von Punktmengen—eine Verallgemeinerung des Längenbegriffs’, Nachr. Ges. Wiss. Göttingen (1914), 404–26.Google Scholar

‡ ‘Dimension und äusseres Mass’, Math. Ann. 79 (1919), 157–79.Google Scholar

§ Rado, T., ‘On the problem of plateau’, Ergebn. Math. Grenzgeb. 2 (1933).Google Scholar

║ Geöcze, Z., ‘Sur l'exemple d'une surface dont l'aire est égale à zéro et qui remplit un cube’, Bull. Soc. Math. France, 41 (1913), 29–31.Google Scholar

† Quart. J. Math. (Oxford Series), 16 (1945), 86–102.Google Scholar

‡ The surface quoted above has an important bearing on the classical problem of the minimum value of the area of the surface of a solid, whose volume has a fixed value. It gives the answer that if the L. -F. definition is adopted, the area may be as small as we please. Since my paper was published it has been pointed out more than once that if the interior volume is constant then the old answer remains valid. Of course, I have never said that any application, past or future, of the L. F. definition has led or would lead necessarily to unexpected results.

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

† Besioovitch, A. S., ‘A general form of the covering principle’, Proc. Cambridge Phil. Soc. 42 (1946), 2.Google Scholar