In Convex Analysis and Nonlinear Optimization by Borwein and Lewis there's the following exercise, stating that a closed approximately convex is convex:

http://s17.postimg.org/fpspftejz/Approximately_Convex_Set.jpg

In (iv) I don't understand why the polar set is a ray. The contradiction in (v) comes from $H$ being a half-space.
$E$ is an euclidean(finite dimensional) normed liniar space. The polar set $H^{\circ}$ is defined as $H^{\circ}=\{\phi\in E\,|\,\langle\phi,x\rangle\leq 1, \forall x\in H\}$.