Consider a pair of tangential curves, one for each consumer as illustrated in Fig. 5, where the point of tangency is shown by the purple dot. Then convexity guarantees that the curves cannot intercept other than at the point of tangency, and the box is accordingly divided into 3 regions. The pale blue area is preferable to the point of tangency for Octavio but worse for Abby; the pale orange area is preferable for Abby but worse for Octavio; and the white area is worse for both. Similar considerations apply to the boundaries. It follows that the point of tangency is Pareto optimal.