Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

We show that an absolutely minimizing function with respect to a convex Hamiltonian H : R(n) ->-> R is uniquely determined by its 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 (a).