Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



A new proof
of Federer's structure theorem
for $k$-dimensional subsets of $\mathbf{R}^{N}$

Author: Brian White
Journal: J. Amer. Math. Soc. 11 (1998), 693-701
MSC (1991): Primary 28A75, 28A78
MathSciNet review: 1603866
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 28A75, 28A78

Retrieve articles in all journals with MSC (1991): 28A75, 28A78

Additional Information

Brian White
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): September 15, 1997
Received by editor(s) in revised form: February 12, 1998
Additional Notes: The author was partially funded by NSF grant DMS-95-04456.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society