The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined.

翻译：半空深度是非对数多变量分析的一个突出工具。 深度的上层组,称为测量的曲面区域, 用作量和量间区域的自然一般化。 最小的非空三角区域, 生成了测量的半空中位, 概括了中位值。 我们关注半空深度的( 反) 射线基点, 这是以覆盖属性来描述半空中位的关键理论结果 。 首先, 在基本测量的最小假设下, 提供了该语句的新的基本证据 。 该证据不仅适用于中位, 也适用于其他三角区域 。 受修正的射线基础定理的技术发展驱动, 我们指定了三角区域、 浮体 和 与深度 额外 equiffine 锥形 之间的连接 。 因此, 获得了深度 严格单调的最小条件 。 深度和 精确估计 的计算 。