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On convex sets and measures

Published online by Cambridge University Press:  24 October 2008

D. G. Larman  and
D. J. Ward
D. G. Larman
Affiliation:
University College London and the University of Sussex
D. J. Ward
Affiliation:
University College London and the University of Sussex
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Extract

If α is a positive real number then, for each set E in R3, we define

where U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 1 , January 1966 , pp. 33 - 42
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

(1)Besicovitch, A. S.Parametric surfaces. II. Proc. Cambridge Philos. Soc. 45 (1949), 14–23.CrossRefGoogle Scholar
(2)Besicovitch, A. S.On surfaces of minimum area. Proc. Cambridge Philos. Soc. 44 (1948), 313–334.CrossRefGoogle Scholar
(3)Ward, D. J.A counter-example in area theory. Proc. Cambridge Philos. Soc. 60 (1964), 821–845.CrossRefGoogle Scholar