  Given an arbitrary closed set A of $\mathbf{R}^{n}$, we establish the
relation between the eigenvalues of the approximate differential of the
spherical image map of A and the principal curvatures of A introduced by
Hug-Last-Weil, thus extending a well known relation for sets of positive reach
by Federer and Zaehle. Then we provide for every $ m = 1, \ldots , n-1 $ an
integral representation for the support measure $ \mu_{m} $ of A with respect
to the m dimensional Hausdoff measure. Moreover a notion of second fundamental
form $Q_{A} $ for an arbitrary closed set A is introduced so that the finite
principal curvatures of A correspond to the eigenvalues of $ Q_{A} $. We prove
that the approximate differential of order 2, introduced in a previous work of
the author, equals in a certain sense the absolutely continuous part of $ Q_{A}
$, thus providing a natural generalization to higher order differentiability of
the classical result of Calderon and Zygmund on the approximate
differentiability of functions of bounded variation.
