Earlier Work
During the 15 years or so years prior to that paper, Federer worked at the technical interface of geometry and measure theory. He focused particularly on surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the analysis of surfaces. His 1947 paper on the rectifiable subsets of n-space characterized purely unrectifiable sets by their "invisibility" under almost all projections. A. S. Besicovitch had proven this for 1-dimensional sets in the plane, but Federer's generalization, valid for subsets of arbitrary dimension in any Euclidean space, was a major technical accomplishment, and later played a key role in Normal and Integral Currents.
In 1958, Federer wrote Curvature Measures, a paper that took some early steps toward understanding second-order properties of surfaces lacking the differentiability properties typically assumed in order to discuss curvature. He also developed and named what he called the coarea formula in that paper. That formula has become a standard analytical tool.
Read more about this topic: Herbert Federer
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