Before revision:  We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well-suited for representing convex functions.
After revision:  Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well-suited for representing convex functions.