  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.
