On singular strictly convex solutions to the Monge-Ampère equation

We show the existence of a strictly convex function $u:B_1\to \mathbb{R}$ with associated Monge-Ampère measure represented by a function $f$ with $0<f<1$ a.e. whose Hessian has a singular part. This extends the work [13] and answers an open question of [14,Sec. 6.2(1)].