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A compact set of disjoint line segments in E3 whose end set has positive measure

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

D. G. Larman
D. G. Larman
Affiliation:
University College London, Gower Street, London, W.C.1.
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Extract

If L is a set of disjoint closed line segments in En, let E(L) denote the end set of L, i.e. the set of end points of members of L.

In [1, 2] V. L. Klee and M. Martin proved the following lemma:

IfLis a disjoint set of closed line segments in E2 such that E(L) is compact, then E(L) has zero 2-measure.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 112 - 125
Copyright
Copyright © University College London 1971

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References

1.Klee, V. L. and Martin, M., “Semi-continuity of the face function of a convex set”, Commentarii Math. Helv.Google Scholar
2.Klee, V. L. and Martin, M., “Must a compact end set have measure zero?” Amer. Math. Monthly 11 (1970), 616–18.CrossRefGoogle Scholar
3.Larman, D. G., “On a conjecture of Klee and Martin for convex bodies”, Proc. London Math. sec., to appear.Google Scholar
4.Bruckner, A. M. and Ceder, J. G., “A note on End-sets” Amer. Math. Monthly, 78 (1971), 516–18.CrossRefGoogle Scholar
5.Davies, Roy O., “On accessibility of plane sets and approximate differentiation of functions of two real variables’, Proc. Camb. Phil. Soc., 48 (1952) 215–232.CrossRefGoogle Scholar
6.Besicovitch, A. S., “On Kakeya's problem and a similar one,” Math. Zeit., 27 (1928), 312–320.CrossRefGoogle Scholar