A property for the Monge-Ampère equation

Let Ω ⊆ ℝn be a non-empty open bounded set and h: Ω → ℝ be a non-negative continuous function. We prove that for any u ∈ C2(Ω) ∩ C1($$\overline{\Omega}$$) solution of the Monge–Ampere equation $${\rm{det}}(D^2u)=h\;\;{\rm{in}}\;\Omega,$$, then ∇u satisfies the convex hull property: ∇u(Ω) ⊆ conv(∇u(∂Ω)).