Halfspace depth ($D(x;P)$)

The halfspace depth of $x\in {\mathbb{R}}^d$ for a probability measure $P$ is the least mass of any closed half-space containing $x$:For an empirical distribution, it is the smallest fraction of observations in such a closed half-space. The boundary is included, so observations exactly on it count.

In one dimension,For observations $0,1,2$, the depth at $1$ is $2/3$, and the depth at either endpoint is $1/3$. Atoms can therefore give depth greater than $1/2$. An invertible affine map preserves halfspace depth, because it carries the family of closed half-spaces bijectively to itself and preserves their associated probabilities. A Tukey median maximizes this depth; the maximizing point need not be unique.