Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals

Abstract

For a closed subset C of a Hilbert space and for a sublinear functional, which is equivalent to the norm, we give conditions guaranteeing existence and uniqueness of the nearest points to C in the sense of the semidistance generated by given sublinear functional. This permits us to construct a continuous retraction onto C well defined in an open neighbourhood of C. In particular, according to one of the conditions, this neighbourhood can be represented in terms of balance between the local strict convexity modulus of the Minkowski (sublinear) functional and the measure of nonconvexity of the set C at each point.