It is a well known result of Caffarelli that an upper and lower bound on the Monge-Amp{\`e}re measure of a convex function u implies the function must actually be strictly convex. A lesser known result, also by Caffarelli, states that if the Monge-Amp{\`e}re of u has only a lower bound, the contact set between u and a supporting affine function must have affine dimension strictly less than n/2. By making a geometric construction involving the subdifferential of a convex function at a singular point, we give an alternative proof of Caffarelli's result. Additionally, this method can be used to extend the result to optimal transport problems with cost functions satisfying the weak Ma-Trudinger-Wang condition. This talk is based on a joint work with Young-Heon Kim.