In this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set. The existence, support size, likelihood equations, and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull. It is shown using geometric techniques that the estimator exists under quite general conditions, with a support size no larger than the number of distinct observations. Analysis of the convex dual of the likelihood set leads to a dual maximization problem. A convergent algorithm is described. The defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures. Sufficient conditions for uniqueness are given. Part II will deal with a special theory for exponential family mixtures.