= Halfspace depth
{title2=$D(x;P)$}
The <halfspace depth> of $x\in\mathbb R^d$ for a <probability measure> $P$ is the least mass of any <closed half-space> containing $x$:
$$
D(x;P)=\inf_{\|u\|=1}P\{z:u^Tz\leq u^Tx\}.
$$
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>,
$$
D(x;P)=\min\{P(X\leq x),P(X\geq x)\}.
$$
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.
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