The affine Bernstein conjecture asks whether an entire smooth locally uniformly convex solution of the affine
maximal equation must be a quadratic polynomial.
Local uniform convexity here means that the Hessian is
positive definite at every point;
a uniform lower bound on all of 
is not assumed.
For the graph of a solution 
, the induced Riemannian metric
is 
.
Euclidean completeness is different from completeness in the affine metric 
.
