  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.
