AI 中文总结
本文以多样性测度为桥梁，探索次模函数、强次模函数与度量理论的联系，定义次模多样性测度并证明其几何嵌入结果，表明次模函数等可由广义外接球半径表示，丰富了相关领域的理论联系。
AI 中文摘要
次模函数及其近亲在组合优化、决策理论和位势理论中发挥关键作用，其部分重要性与实用性源于它们与凸函数和多面体的联系。本文探索这些函数与度量理论之间的联系，该联系由多样性测度（一种近期发展的、从仅针对点对扩展至（有限）集合的度量空间推广概念）作为桥梁。次模函数和强次模函数均对应自然的多样性测度类；本文定义的次模多样性测度本质上是单调不减、相交次模且在单点集上取值为零的函数。我们证明了这些多样性测度的新几何嵌入结果，尤其表明次模函数、强次模函数和XOS函数可由广义外接球半径表示，广义外接球半径是凸分析中的集合函数，等于给定凸体覆盖一组点所需拉伸的量。
英文摘要
Submodular functions and their close relatives play a key role in combinatorial optimization, decision theory and potential theory. Part of their importance and usefulness stems from the connections with convex functions and polytopes. Here we explore connections between these functions and metric theory, with the bridge provided by diversities, a recently developed generalization of metric spaces to (finite) sets rather than just pairs. Both submodular functions and strongly submodular functions correspond to natural classes of diversities. Submodular diversities, as we define them here, are essentially non-decreasing, intersecting submodular functions which vanish on singletons. We prove new geometric embedding results for these diversities. In particular we show that submodular, strongly submodular, and XOS functions can be represented by the generalized circumradius, a set function in convex analysis equal to the amount a given convex body needs to be stretched to cover a set of points.
