论文标题
锥形和决策理论的应用的表示
Representations of cones and applications to decision theory
论文作者
论文摘要
让$ c $为本地凸出的hausdorff拓扑矢量空间$ x $，其中包含$ 0 $。我们表明，存在一个（本质上是独特的）非空的家庭$ \ mathscr {k} $的拓扑二$ x^\ prime的nonepty子集，使得$ c = \ {x \ in x：\ forall k \ in x：\ forall k \ in \ mathscr {k}，\ in k {k}，\ in k in k in k in k in k in k in k in k in k in k in k in k in k in k，ge 0.然后，我们确定了家庭$ \ mathscr {k} $的其他属性，这些属性是封闭的凸锥，开放式凸锥，封闭锥和凸锥的特征。例如，如果$ x $是Banach的空间，则$ C $是一个封闭的圆锥，并且仅当家庭$ \ Mathscr {k} $才能使用非空置凸形套件选择。 这些表示形式提供了决策理论最新结果的抽象版本，并为我们提供了获得新框架的适当框架。这使我们能够表征满足某些概率措施的独立性公理的预订，并在[Conatemetrica〜 \ textbf {87}（2019）中回答了一个空旷的问题，否。 3，933--980]。
Let $C$ be a cone in a locally convex Hausdorff topological vector space $X$ containing $0$. We show that there exists a (essentially unique) nonempty family $\mathscr{K}$ of nonempty subsets of the topological dual $X^\prime$ such that $$ C=\{x \in X: \forall K \in \mathscr{K}, \exists f \in K, \,\, f(x) \ge 0\}. $$ Then, we identify the additional properties on the family $\mathscr{K}$ which characterize, among others, closed convex cones, open convex cones, closed cones, and convex cones. For instance, if $X$ is a Banach space, then $C$ is a closed cone if and only if the family $\mathscr{K}$ can be chosen with nonempty convex compact sets. These representations provide abstract versions of several recent results in decision theory and give us the proper framework to obtain new ones. This allows us to characterize preorders which satisfy the independence axiom over certain probability measures, answering an open question in [Econometrica~\textbf{87} (2019), no. 3, 933--980].
