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A counter-example in area theory

Published online by Cambridge University Press:  24 October 2008

D. J. Ward  and
E. R. Reifenberg
D. J. Ward
Affiliation:
University of Bristol
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Extract

In area theory, Hausdorff measure has been mainly used in the form of Hausdorff ‘spherical’ measure (l) in spite of the fact that work on the geometry of sets of points has mainly used Hausdorff ‘convex’ measure (2,4). The reason for this lies in the inequality This inequality is known (1,3) when the measures are interpreted aslspherical’. But as has for some time been conjectured and this paper shows, with the proper normalization it is false for the convex case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 821 - 845
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Besicovitch, A. S.Parametric Surfaces. II. Lower semicontinuity. Proc. Cambridge Philos. Soc. 45 (1949), 14–23.CrossRefGoogle Scholar
(2)Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points, (II). Math. Ann. 115 (1938), 296–329.CrossRefGoogle Scholar
(3)Eilenberg, S. and Harrold, O. G., JR. Continua of finite linear measure. American J. Math. 65 (1943), 141.CrossRefGoogle Scholar
(4)Marstrand, J. M.Hausdorff two-dimensional measure in three-space. Proc. London Math. Soc. 11 (1961), 91–108.CrossRefGoogle Scholar
(5)Reifenberg, E. R.On the tangential properties of surfaces. Bull. American Math. Soc. 68 (1952), 213–216.CrossRefGoogle Scholar
(6)Reifenberg, E. R.Solution of the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math. 104 (1960), 1–92.CrossRefGoogle Scholar