For a bounded domain, we consider the L^\infty-functional involving a nonnegative Hamilton function. Under the continuous Dirichlet boundary condition and some assumptions of Hamiltonian H, the uniqueness of absolute minimizers for Hamiltonian H is established. This extendes the uniqueness theorem to a larger class of Hamiltonian $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by Jensen-Wang-Yu. Our proofs rely on geometric structure of the action function induced by Hamiltonian H(x,p), and the identification of the absolute subminimality with convexity of the associated Hamilton-Jacobi flow.