  Let $u$ minimize the functional $F(u) = \int_\Omega f(\nabla u(x))\, dx$ in
the class of convex functions $u : \Omega \to {\mathbb R}$ satisfying $0 \le u
\le M$, where $\Omega \subset {\mathbb R}^2$ is a compact convex domain with
nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a
$C^2$ function, with $\{ \xi : \, \text{the smallest eigenvalue of} \, f"(\xi)
\, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let
epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of
epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$.
As a consequence, an optimal function $u$ is uniquely defined by the set of
singular points of epi$(u)$. This result is applicable to the classical
Newton's problem, where $F(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{-1}\, dx$.
