Sets that maximize probability and a related variational problem

Abstract

Let  be a random variable of a Riemannian manifold. We assume that the C 2 -probability density function of  exists. This research addresses two variational questions. The first concerns sets that maximize their probability among those that have a fixed volume. We prove that such a set must have a probability density function that is constant along its boundary; equivalently, such a set must be a density level set. We also obtain the equations related to the maximization property (the stability of the solutions). The other variational problem is the inverse of the first question, namely which sets minimize their volume among those sets which have a predetermined probability? The solution of this problem will define a notion of a quantile set. We show that the solutions of both variational problems coincide (the critical point equation and the stability condition). As theoretical applications, we consider a decision-making problem and fuzzy sets. As practical applications, we first explore how to locate a powerplant, and subsequently develop a model for a distribution of cheetahs.