We say a measure is @math @math -rectifiable if there is a countable union of @math @math -surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in @math to be @math @math -rectifiable, with @math . We assume a priori the measure to have positive and finite upper density. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the @math constants depend on such a function. Along the way we also give sufficient conditions for @math parametrizations for one-sided Reifenberg flat sets in terms of the same square function. Key tools for the proof come from Guy David and Tatiana Toro's Reifenberg parametrizations of sets with holes in the Holder and Lipschitz categories.
@cite_52: In this paper we prove finiteness principles for ( C^m ( R ^n , R ^D) ) and ( C^ m-1,1 ( R ^n, R ^D) ) selections. In particular, we provide a proof for a conjecture of Brudnyi-Shvartsman (1994) on Lipschitz selections for the case when the domain is ( X= R ^n ).
In the last few years, C. Fefferman, A. Israel, and G.K. Luli @cite_52 have been investigating Whitney type extension problems for @math maps, finding conditions to fit smooth functions to data.