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

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.