A new proof

of Federer's structure theorem

for -dimensional subsets of

Author:
Brian White

Journal:
J. Amer. Math. Soc. **11** (1998), 693-701

MSC (1991):
Primary 28A75, 28A78

MathSciNet review:
1603866

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that Federer's structure theorem for -dimensional sets in follows from the special case of -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. J. Falconer,*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284****[Fe1]**Herbert Federer,*The (𝜙,𝑘) rectifiable subsets of 𝑛-space*, Trans. Amer. Soc.**62**(1947), 114–192. MR**0022594**, 10.1090/S0002-9947-1947-0022594-3**[Fe2]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[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]**Pertti Mattila,*Geometry of sets and measures in Euclidean spaces*, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR**1333890****[S]**Leon Simon,*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417**

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Additional Information

**Brian White**

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

Email:
white@math.stanford.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-98-00267-7

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