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On the definition of tangents to sets of infinite linear measure†

Published online by Cambridge University Press:  24 October 2008

A. S. Besicovitch
A. S. Besicovitch
Affiliation:
Trinity CollegeCambridge
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Extract

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 1 , January 1956 , pp. 20 - 29
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

(1)Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points. I. Math. Ann. 98 (1928), 422–64.CrossRefGoogle Scholar
(2)Besicovitch, A. S.Concentrated and rarified sets of points. Acta math., Stockh., 62 (1934), 289–300.CrossRefGoogle Scholar
(3)Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points. II. Math. Ann. 115 (1938), 296–329.CrossRefGoogle Scholar
(4)Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points. III. Math. Ann. 116 (1939), 349–57.CrossRefGoogle Scholar
(5)Besicovitch, A. S.On existence of subsets of finite measure of sets of infinite measure. Indag. math. 14 (1952), 339–44.CrossRefGoogle Scholar
(6)Davees, R. O. J.Lond. math. Soc. (to be published shortly).Google Scholar