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On the set of distances between points of a general metric space

Published online by Cambridge University Press:  24 October 2008

A. S. Besicovitch  and
S. J. Taylor
A. S. Besicovitch
Affiliation:
Trinity CollegeCambridge
S. J. Taylor
Affiliation:
PeterhouseCambridge
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Extract

The well-known Steinhaus theorem (2) with respect to the set of distances of linear sets of positive Lebesgue measure has been generalized to the case of linearly measurable subsets of rectifiable curves in the Euclidean plane by Besicovitch and Miller (1). An extension of the theorem to rectifiable curves in Euclidean n-space is immediate. Prof. A. P. Morse has suggested the problem as to whether or not the theorem still remains true in a general metric space. By defining a particular curve ℒ in a metric space we prove that the answer to this question is in the negative.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 2 , April 1952 , pp. 209 - 214
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Besicovitch, A. S. and Miller, D. S.On the set of distances between the points of a Caratheodory linearly measurable plane set. Proc. Lond. math. Soc. (2), 50 (1948), 305–16.CrossRefGoogle Scholar
(2)Steinhaus, H.Sur les distances des points des ensembles de mesure positive. Fundam. Math. 1 (1920), 93–104.CrossRefGoogle Scholar