Abstract

For a closed subset CC of a Hilbert space (H,∥⋅∥)\left( H,\left\Vert \cdot \right\Vert \right) and for a sublinear functional ρ:H→R+\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 CC in the sense of the semidistance generated by ρ\rho. This permits us to construct a continuous retraction onto CC well defined in a neighbourhood U⊃C\mathcal{U}\supset C. In particular, according to one of the conditions, U\mathcal{U} can be represented in terms of balance between the local strict convexity modulus of ρ\rho and the measure of nonconvexity of the set CC at each point.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Time-minimum problem
  • Minkowski functional
  • generalized projection
  • strict convexity
  • curvature
  • proximal normals

Mathematics Subject Classification

49J52, 49N15

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Suggested citation

V. V. Goncharov, F. F. Pereira. “Neighbourhood Retractions of Nonconvex Sets in a Hilbert Space via Sublinear Functionals.” Journal of Convex Analysis 18 (2011), No. 1, 1–36. https://doi.org/10.68381/jca18001

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.