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SMOOTH NORMAL APPROXIMATIONS OF EPI-LIPSCHITZIAN SUBSETS OF Rn∗

A sequence (Mk) of closed subsets of Rn converges normally to M ⊂ Rn if (sc) M = lim supMk = lim inf Mk in the sense of Painlevé–Kuratowski and (nc) lim sup G(NMk ) ⊂ G(NM ), where G(NM ) (resp., G(NMk )) denotes the graph of NM (resp., NMk ), Clarke’s normal cone to M (resp., Mk). This paper studies the normal convergence of subsets of Rn and mainly shows two results. The first result states that every closed epi-Lipschitzian subset M of Rn, with a compact boundary, can be approximated by a sequence of smooth sets (Mk), which converges normally to M and such that the sets Mk and M are lipeomorphic for every k (i.e., the homeomorphism between M and Mk and its inverse are both Lipschitzian). The second result shows that, if a sequence (Mk) of closed subsets of Rn converges normally to an epi-Lipschitzian set M , and if we additionally assume that the boundary of Mk remains in a fixed compact set, then, for k large enough, the sets Mk and M are lipeomorphic. In Cornet and Czarnecki [Cahier Eco-Maths 95-55, 1995], direct applications of these results are given to the study (existence, stability, etc.) of the generalized equation 0 ∈ f(x∗) + NM (x∗) when M is a compact epi-Lipschitzian subset of Rn and f : M → Rn is a continuous map (or more generally a correspondence).