Let $𝒳$ be a random variable of a Riemannian manifold. We assume that the C²-probability density function of $𝒳$ 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.

中文翻译：

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最大化概率的集合和相关的变分问题

让 $𝒳$是黎曼流形的随机变量。我们假设C ^{2 -}概率密度函数为$𝒳$存在。这项研究解决了两个不同的问题。第一个涉及在具有固定体积的集合中最大化其概率的集合。我们证明这样的集合必须具有沿其边界恒定的概率密度函数；等价地，这样的集合必须是密度水平集合。我们还获得了与最大化属性（解的稳定性）相关的方程。另一个变分问题是第一个问题的逆问题，即在那些具有预定概率的集合中，哪些集合的体积最小？这个问题的解决方案将定义分位数集的概念。我们表明两个变分问题的解是一致的（临界点方程和稳定性条件）。作为理论应用，我们考虑决策问题和模糊集。