To begin, Chebyshev Distance is defined as $d(a_x, a_y, b_x, b_y) = \max\left(| a_x - a_y |, | b_x - b_y | \right)$ in $\mathbb{R}^2$. This is the equivalent of the $L^\infty$ norm. The question is: given $N$ points in $\mathbb{R}^2$, how do we find the point(s) in $\mathbb{R}^2$ that minimize the maximum Chebyshev distance to these points?

Formalization

Given $N$ points $(x_i, y_i)$ for $i = 1, 2, \ldots, N$, we want to minimize function $f$ defined as $f(x, y) = \max_{i=1}^N \max\left( |x - x_i|, |y - y_i| \right)$. Simple enough, the solution is to find the minimum axis-aligned bounding box that contains all the points, which is a rectangle with the left top corner as $(x_{min}, y_{min})$ and the right bottom corner as $(x_{max}, y_{max})$. The minimum Chebyshev distance is then

achieved at the center of the bounding box.

Trivia

Chebyshev distance was introduced in an example involving chess.