
Journal of Convex Analysis 18 (2011), No. 1, 001036 Copyright Heldermann Verlag 2011 Neighbourhood Retractions of Nonconvex Sets in a Hilbert Space via Sublinear Functionals Vladimir V. Goncharov CIMAUE, Dep. de Matemática, Universidade de Évora, Rua Romăo Ramalho 59, 7000671 Évora, Portugal goncha@uevora.pt Fátima F. Pereira CIMAUE, Dep. de Matemática, Universidade de Évora, Rua Romăo Ramalho 59, 7000671 Évora, Portugal fmfp@uevora.pt [Abstractpdf] For a closed subset $C$\ of a Hilbert space $\left( H,\left\Vert \cdot \right\Vert \right) $ and for a sublinear functional $\rho :H\rightarrow \mathbb{R}^{+}$, which is equivalent to the norm $\left\Vert \cdot \right\Vert $, we give conditions guaranteeing existence and uniqueness of the nearest points to $C$ in the sense of the semidistance generated by $% \rho $. This permits us to construct a continuous retraction onto $C$ \ well defined in a neighbourhood\ $\mathcal{U}\supset C$. In particular, according to one of the conditions, $\mathcal{U}$\ can be represented in terms of balance between the local strict convexity modulus of $\rho $ and the measure of nonconvexity of the set $C$ at each point. Keywords: Timeminimum problem, Minkowski functional, generalized projection, strict convexity, curvature, proximal normals. MSC: 49J52, 49N15 [ Fulltextpdf (307 KB)] for subscribers only. 