Distances between sets arise naturally when modeling stochastic dependence on collections of spatial supports, including settings with point-referenced and areal observations. 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. By Schoenberg's theorem, this implies an isometric embedding into a Hilbert space and guarantees the validity of broad classes of isotropic covariance functions, including the Matérn and powered exponential families, for set-indexed random fields. The construction reduces dependence between sets to low-dimensional geometric summaries, leading to substantial simplifications in covariance evaluation.

翻译：集合间距离在建模空间支撑集（包括点参考观测和区域观测）的随机依赖性时自然产生。然而，常用的集合距离构造（包括由豪斯多夫距离导出的距离）通常不具备条件负定性，从而无法用于各向同性协方差模型。我们提出球-豪斯多夫距离，其定义为度量空间中有界集的最小包围球之间的豪斯多夫距离。对于长度空间，我们基于相关球心和半径推导出该距离的显式表示。我们证明，当底层度量具有条件负定性时，球-豪斯多夫距离同样具有条件负定性。根据Schoenberg定理，这意味着可等距嵌入希尔伯特空间，并保证广泛类型的各向同性协方差函数（包括Matérn族和幂指数族）对集合索引随机场的有效性。该构造将集合间依赖性简化为低维几何摘要，从而显著简化协方差计算。