Before revision:  In particular, we prove that ReLU nets with width d+1 can approximate any continuous convex function of d variables arbitrarily well . Moreover, when approximating convex, piecewise affine functions by such nets, we obtain matching upper and lower bounds on the required depth, proving that our construction is essentially optimal .
After revision:  In particular, we prove that ReLU nets with width d+1 can approximate any continuous convex function of d variables arbitrarily well .