Before revision:  Moreover, when approximating convex, piecewise affine functions by width d+1 ReLU nets, we obtain matching upper and lower bounds on the required depth, proving that our construction is essentially optimal.  ]
After revision:  Moreover, when approximating convex, piecewise affine functions by width d+1 ReLU nets, we obtain matching upper and lower bounds on the required depth, proving that our construction is essentially optimal. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube ]