Convex extension preserving Lipschitz constant (Q1364821)

The author obtains necessary and sufficient conditions for the convex extension of functions. Precisely, let \(\Omega\subset {\mathbb R}^{n}\) be a non-empty bounded, open and convex set and let \(f\) be a real-valued function defined on the boundary \(\partial \Omega.\) Then, \(f\) admits a convex extension \(w_{f}:\) \({\mathbb R}^{n}\to{\mathbb R}\) satisfying the Lipschitz condition with a constant \(L\) if and only if \[ f(z)-{ {f(x)+f(y)}\over {2}} \leq L\left\| z-{{x+y}\over {2}}\right\| \tag{\(*\)} \] for all \(x, y, z\in \partial \Omega.\) As consequence, it follows that the solution \(u\) of the Dirichlet problem for the degenerate Monge-Ampère equation \[ \text{det}\left({{\partial^{2}u}\over {\partial x_{i}\partial x_{j}}}\right)=0\;\text{ in} \Omega,\qquad u=f\;\text{ on} \partial \Omega, \] is Lipschitz continuous if and only if \(f\) satisfies \((*)\).