Abstract
A class of subsets of ℝd which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d − 1curvature measures) is introduced as a (d − 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.
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Rataj, J., Zähle, M. Curvatures and Currents for Unions of Sets with Positive Reach, II. Annals of Global Analysis and Geometry 20, 1–21 (2001). https://doi.org/10.1023/A:1010624214933
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DOI: https://doi.org/10.1023/A:1010624214933