  We show that absolutely minimizing functions relative to a convex Hamiltonian
$H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary
values under minimal assumptions on $H.$ Along the way, we extend the known
equivalences between comparison with cones, convexity criteria, and absolutely
minimizing properties, to this generality. These results perfect a long
development in the uniqueness/existence theory of the archetypal problem of the
calculus of variations in $L^\infty.$
