Journal of the American Mathematical Society

A new proof of Federer's structure theorem for $k$

-dimensional subsets of $\mathbf{R}^{N}$

Author(s): Brian White
Journal: J. Amer. Math. Soc. 11 (1998), 693-701.
MSC (1991): Primary 28A75, 28A78
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Abstract: We prove that Federer's structure theorem for $k$

-dimensional sets in $\mathbf{R}^{N}$ follows from the special case of $1$

-dimensional sets in the plane, which was proved earlier by Besicovitch.

[B]: A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points I, Math. Ann. 98 (1928), 422-464; II, Math. Ann 115 (1938), 296-329; III, Math. Ann 116 (1939), 349-357.
[Fa]: K. Falconer, The geometry of fractal sets, Cambridge U. Press, 1985. MR 88d:28001
[Fe1]: H. Federer, The $(\phi ,k)$ -rectifiable subsets of $n$ space, Trans. Amer. Math. Soc. 62 (1947), 114-192. MR 9:231c
[Fe2]: H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. MR 41:1976
[J]: P. Jones, N. Katz, and A. Vargas, Checkerboards, Lipschitz functions and uniform rectifiability, Rev. Mat. Iberoamericana 13 (1997), 189-210. CMP 97:16
[M]: P. Mattila, Geometry of sets and measures in euclidean spaces: fractals and rectifiability, Cambridge U. Press, 1995. MR 96h:28006
[S]: L. Simon, Lectures on geometric measure theory, Australian National Univ., Canberra, 1983. MR 87a:49001

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