Distances between sets arise naturally in data fusion problems involving both point-referenced and areal observations, as well as in set-indexed stochastic processes more broadly. However, commonly used constructions of distances on sets, including those derived from the Hausdorff distance, generally fail to be conditionally negative definite, precluding their use in isotropic covariance models. We propose the ball-Hausdorff distance, defined as the Hausdorff distance between the minimum enclosing balls of bounded sets in a metric space. For length spaces, we derive an explicit representation of this distance in terms of the associated centers and radii. We show that the ball-Hausdorff distance is conditionally negative definite whenever the underlying metric is conditionally negative definite, which implies, via Schoenberg's theorem, an isometric embedding into a Hilbert space. As a consequence, broad classes of isotropic covariance functions, including the Matérn and powered exponential families, are valid for random fields indexed by sets. The resulting construction reduces set-to-set dependence to low-dimensional geometric summaries, leading to substantial computational simplifications in covariance evaluation.

翻译：集合间距离在涉及点参考和区域观测的数据融合问题中自然产生，在更广泛的集合索引随机过程中亦是如此。然而，常用的集合距离构造方法（包括源自豪斯多夫距离的方法）通常无法满足条件负定性，从而阻碍了其在各向同性协方差模型中的应用。我们提出了球-豪斯多夫距离，其定义为度量空间中有界集合的最小包围球之间的豪斯多夫距离。对于长度空间，我们基于相关中心和半径推导了该距离的显式表示。我们证明，当基础度量满足条件负定性时，球-豪斯多夫距离也具有条件负定性，这通过Schoenberg定理意味着可等距嵌入希尔伯特空间。因此，包括Matérn族和幂指数族在内的广泛类别的各向同性协方差函数对于集合索引的随机场是有效的。该构造将集合间依赖性简化为低维几何摘要，从而在协方差计算中实现了显著的简化。