A geometric second-order-rectifiable stratification for closed subsets of Euclidean space

Abstract

Defining the $m$-th stratum of a closed subset of an $n$ dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least $n-m$ linearly independent directions, we establish that the $m$-th stratum is second-order rectifiable of dimension $m$ and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a new criterion for second-order rectifiability.

13 pages. v2: corrected a typographical error in the introduction, and updated the affiliation of the first author

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