We investigate the asymptotics for two geometric measures, geometric quantiles and halfspace depths. While much literature is known on the population side, we fill out some gaps there to obtain a full picture, before turning to the sample versions, where the questions on asymptotics become crucial in view of applications. This is the core of the paper: We provide rates of convergence for the sample versions and address the extremal behaviour of the geometric measures according to the type of underlying distribution.

翻译：我们研究了两种几何度量——几何分位数与半空间深度的渐近性质。尽管关于总体分布的文献已有较多探讨，但为了获得完整的理论图景，我们首先填补了其中的若干空白，随后转向样本版本——由于实际应用的需求，样本版本的渐近问题显得至关重要。这正是本文的核心：我们给出了样本版本的收敛速率，并根据基础分布的类型，探讨了这些几何度量的极值行为。