Before revision:  We prove that ReLU nets with width 2d+2 can approximate any continuous scalar function on the d-dimensional cube [0,1]^d arbitrarily well. We obtain quantitative depth estimates for such approximations .
After revision:  We prove that ReLU nets with width 2d+2 can approximate any continuous scalar function on the d-dimensional cube [0,1]^d aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions .