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

## TL;DR

This paper introduces a new geometric framework for stratifying closed subsets of Euclidean space, proving that each stratum is second-order rectifiable and a Borel set, extending known results beyond convex sets.

## Contribution

It establishes that the $m$-th stratum of a closed set is second-order rectifiable and Borel, providing a new criterion that generalizes previous results for convex sets and sets of positive reach.

## Key findings

- Each stratum is second-order rectifiable of dimension m.
- The m-th stratum is a Borel set.
- The result extends known properties from convex sets to more general closed sets.

## 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.