Neighbourhood Retractions of Nonconvex Sets in a Hilbert Space via Sublinear Functionals

V. V. Goncharov ; F. F. Pereira

Journal of convex analysis, Tome 18 (2011) no. 1, pp. 1-36

Résumé

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.

Détail

Classification : 49J52, 49N15
Mots-clés : Time-minimum problem, Minkowski functional, generalized projection, strict convexity, curvature, proximal normals