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

For a closed subset

of a Hilbert space

\left( H,\left\Vert \cdot \right\Vert \right) ( H , ∥ ⋅ ∥ )

and for a sublinear functional

\rho:H\rightarrow \mathbb{R}^{+} ρ : H → 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 C

in the sense of the semidistance generated by

\rho ρ

. This permits us to construct a continuous retraction onto

C C

well defined in a neighbourhood

\mathcal{U}\supset C U ⊃ C

. In particular, according to one of the conditions,

\mathcal{U} U

can be represented in terms of balance between the local strict convexity modulus of

\rho ρ

and the measure of nonconvexity of the set

C C

at each point.