  We study the question whether Lipschitz minimizers of $\int F(\nabla u)\,dx$
in $\mathbb{R}^n$ are $C^1$ when $F$ is strictly convex. Building on work of De
Silva-Savin, we confirm the $C^1$ regularity when $D^2F$ is positive and
bounded away from finitely many points that lie in a $2$-plane. We then
construct a counterexample in $\mathbb{R}^4$, where $F$ is strictly convex but
$D^2F$ degenerates on the intersection of a Simons cone with $S^3$. Finally we
highlight a connection between the case $n = 3$ and a result of Alexandrov in
classical differential geometry, and we make a conjecture about this case.
