Before revision:  The usual way to get uniqueness is to assume the function is strictly quasiconvex and the domain is convex. Outside of a few special cases, verifying uniqueness without convexity has not been done and is often just assumed. The main result of this paper establishes uniqueness without assuming convexity by relying on an easy-to-verify nondegeneracy condition. The main result of this paper has widespread application beyond econometrics. Six applications are discussed : uniqueness of M-estimators, utility maximization with a nonconvex budget set, uniqueness of the policy function in dynamic programming, envelope theorems, limit theory in weakly identified models, and functionals of Brownian motion .
After revision:  This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed : uniqueness of M-estimators, utility maximization with a nonconvex budget set, uniqueness of the policy function in dynamic programming, envelope theorems, limit theory in weakly identified models, and functionals of Brownian motion .