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The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space

Published online by Cambridge University Press:  26 February 2010

G. Ewald ,
D. G. Larman  and
C. A. Rogers
G. Ewald
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
D. G. Larman
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
C. A. Rogers
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
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Extract

It is well-known and easy to prove that the maximal line segments on the boundary of a convex domain in the plane are countable. T. J. McMinn [1] has shown that the end-points of the unit vectors drawn from the origin in the directions of the line segments lying on the surface of a convex body in 3-dimensional Euclidean space E3 form a set of σ-finite linear Hausdorff measure on the 2-dimensional surface of the unit ball. A. S. Besicovitch [2] has given a simpler proof of McMinn's result. W. D. Pepe, in a paper to appear in the Proc. Amer. Math. Soc., has extended the result to E4. In this paper we generalize McMinn's result to En by use of Besicovitch's method, proving:

THEOREM 1. If K is a convex body in En, the set S, of end-points of the vectors drawn from origin in the directions of the line segments lying on the surface of K, is a set of σ-finite (n − 2)-dimensional Hausdorff measure on the (n − 1)-dimensional surface of the unit ball.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 1 - 20
Copyright
Copyright © University College London 1970

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References

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