Title: Halfspace depth and floating bodies: Geometry in multivariate statistics.

Abstract:

Statistical depth is a non-parametric tool applicable to multivariate and non-Euclidean data, whose main goal is a reasonable generalisation of quantiles to multivariate and more exotic datasets. We discuss the halfspace depth, arguably the most important depth in statistics. That depth was first proposed in 1975; its rigorous investigation starts in the 1990s, and still an abundance of open problems stimulates the research in the area. We present interesting links of the halfspace depth with well-studied concepts from geometry. Using these relations we partially resolve several open problems concerning the depth, and outline perspectives for future research both in the area of depth and in convex geometry.  The talk is intended to be largely self-contained; no particular knowledge of probability, statistics, or geometry is necessary.