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Abstract:
The first main goal of this thesis is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for every integer k ≥ 2 we introduce the approximate differential of order k of A and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications.
The other goal is to extend to general closed sets a well known theorem of Alberti on the second order rectifiability properties of the boundary of convex bodies. The Alberti theorem provides a stratification of second order rectifiable subsets of the boundary of a convex body based on the dimension of the (convex) normal cone. Considering a suitable generalization of this normal cone for general closed subsets of the Euclidean space and employing some results from the first part we can prove that the same stratification exists for every closed set.
