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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

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