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.
翻译:我们研究以自然的轴心界定的深度测量系列。 我们发现任何这种深度测量都是Tukey深度的常数系数近似值。 我们进一步调查了深度区域的方方面面, 发现卡莱为特维尔贝格深度引入的Cascade猜想, 所有深度测量都符合我们最严格的轴心, 包括Tukey深度。 沿着这条路, 我们引入并研究一种新的深度测量, 叫做包含深度, 我们认为这是独立的兴趣, 并显示它与某些两色点集的 持续折射 Radon 定理的关联 。