Approximative properties of sets with the convex complement

Approximative properties of sets with the convex complement are studied. In Hubert space, a closed set is constructed which has the convex bounded complement and whose distance function is Gâteaux differentiable at each point of the complement. Examples of closed antiproximinal sets with the convex bounded complement are given in the spaces $C(Q)$, $L_{\infty}[S,\Sigma,\mu]$, $L_1[s,\Sigma,\mu]$.