Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
HTML articles powered by AMS MathViewer
- by Bernd Kirchheim PDF
- Proc. Amer. Math. Soc. 121 (1994), 113-123 Request permission
Abstract:
We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the existence of a metric differential of Lipschitzian functions also in situations where the well-known theorem of Rademacher fails.References
- N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147–190. MR 425608, DOI 10.4064/sm-57-2-147-190 S. Banach and S. Mazur, Zur Theorie der linearen Dimension, Studia Math. 4 (1933), 100-112.
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. SosinskiÄ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- William B. Johnson, Joram Lindenstrauss, and Gideon Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), no. 2, 129–138. MR 852474, DOI 10.1007/BF02764938 A. Kolmogoroff, Beiträge zur MaĂźtheorie, Math. Ann. 107 (1932), 351-366. J. Lukeš, J. MalĂ˝, and L. ZajĂÄŤek, Fine topological methods in real analysis and potential theory, Lecture Notes in Math., vol. 1189, Springer, New York, 1986.
- Pertti Mattila, Hausdorff $m$ regular and rectifiable sets in $n$-space, Trans. Amer. Math. Soc. 205 (1975), 263–274. MR 357741, DOI 10.1090/S0002-9947-1975-0357741-4
- Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454 D. Preiss and J. Tišer, On Besicovitch $\frac {1}{2}$-problem, J. London Math. Soc. (2) 45 (1992), 279-287.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 113-123
- MSC: Primary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-1994-1189747-7
- MathSciNet review: 1189747