Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Journal of Convex Analysis 18 (2011), No. 1, 001--036Copyright Heldermann Verlag 2011 Neighbourhood Retractions of Nonconvex Sets in a Hilbert Space via Sublinear Functionals Vladimir V. Goncharov CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal goncha@uevora.pt Fátima F. Pereira CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal fmfp@uevora.pt [Abstract-pdf] 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: Time-minimum problem, Minkowski functional, generalized projection, strict convexity, curvature, proximal normals. MSC: 49J52, 49N15 [ Fulltext-pdf  (307  KB)] for subscribers only.