围封深度及其他深度量度

摘要：

我们研究家庭的深度措施所定义的自然公理集。我们证明，任何这样的深度测量都是Tukey深度的常数因子近似。我们进一步调查的深度区域的尺寸，表明级联猜想，介绍了卡莱Tverberg深度，持有所有的深度措施，满足我们最严格的公理集，其中包括Tukey深度。一路上，我们介绍和研究一个新的深度测量封闭深度，我们相信是独立的利益，并显示其关系到一个常数分数Radon定理在某些双色点集。

英文摘要：

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

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