Abstract
Statistical depth is a non-parametric tool applicable to multivariate and non-Euclidean data, whose goal is a reasonable generalisation of quantiles to multivariate and more exotic datasets. We discuss the halfspace depth, the most classical depth in statistics. J. W. Tukey proposed that depth in 1975; its rigorous investigation started in the 1990s, and still, an abundance of open problems stimulates research in the area.
We present surprising links of the halfspace depth with well-studied concepts from convex geometry. Using these relations, we partially resolve several open problems concerning depth. In particular, we resolve the 30-year-old characterisation conjecture, asking whether two different measures can correspond to the same halfspace depth.
