Journal of Convex Analysis 18 (2011), No. 1, 001--036
Copyright 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 Romăo Ramalho 59, 7000-671 Évora, Portugal
goncha@uevora.pt
Fátima F. Pereira
CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romăo 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
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