  Two different proofs for an inf-sup type representation formula (minimax
formula) of the additive eigenvalues corresponding to first-order
Hamilton-Jacobi equations are given for quasiconvex (level-set convex)
Hamiltonians not necessarily convex. The first proof, which is similar to known
proofs for convex Hamiltonians, invokes a Jensen-like inequality for
quasiconvex functions instead of the standard Jensen's inequality. The second
proof is completely different with elementary calculations. It is based on
convergence of derivatives of mollified Lipschitz continuous functions whose
proof is also given. These methods also relate to an approximation problem of
viscosity solutions.
