Tukey depth

In statistics and computational geometry, the Tukey depth is a measure of the depth of a point in a fixed set of points. The concept is named after its inventor, John Tukey. Given a set of n points $$\mathcal{X}_n = \{X_1,\dots,X_n\}$$ in d-dimensional space, Tukey's depth of a point x is the smallest fraction (or number) of points in any closed halfspace that contains x.

Tukey's depth measures how extreme a point is with respect to a point cloud. It is used to define the bagplot, a bivariate generalization of the boxplot.

For example, for any extreme point of the convex hull there is always a (closed) halfspace that contains only that point, and hence its Tukey depth as a fraction is 1/n.

Definitions
Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud $$\mathcal{X}_n$$, is defined as

$$ D(x;\mathcal{X}_n) = \inf_{v\in\mathbb{R}^d, \|v \|=1} \frac{1}{n}\sum_{i=1}^n \mathbf{1}\{ v^T (X_i - x) \ge 0\}, $$

where $$\mathbf{1}\{\cdot\}$$ is the indicator function that equals 1 if its argument holds true or 0 otherwise.

Population Tukey's depth of x wrt to a distribution $$P_X$$ is

$$ D(x; P_X) = \inf_{v\in\mathbb{R}^d, \|v \|=1} P(v^T (X - x) \ge 0), $$

where X is a random variable following distribution $$P_X$$.

Tukey mean and relation to centerpoint
A centerpoint c of a point set of size n is nothing else but a point of Tukey depth of at least n/(d + 1).