Approximate properties of weakly convex sets in spaces with nonsymmetric seminorm

We consider the support conditions of strong and weak convexity with respect to a nonsymmetric seminorm. We obtain the theorem about the diameter of the epsilon-projection onto a set which satisfies the support condition of weak convexity and the existence and uniqueness theorem for the projection of a point from the Chebyshev layer of such a set. We prove the theorem which states that for two sets which satisfy the support condition of strong and weak convexity correspondingly and are sufficiently close to each other there exists a pair of nearest (in the sense of the nonsymmetric seminorm) points.