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

${\displaystyle D(x;{\mathcal {X}}_{n})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}{\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} \{v^{T}(X_{i}-x)\geq 0\},}$ 

where ${\displaystyle \mathbf {1} \{\cdot \}}$  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 ${\displaystyle P_{X}}$  is

${\displaystyle D(x;P_{X})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}P(v^{T}(X-x)\geq 0),}$ 

where X is a random variable following distribution ${\displaystyle P_{X}}$ .

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).

• Centerpoint (geometry)