Suppose S is a finite set of points in the plane, and the distance between two points s and q in the plane is measured by the L_p distance L_p(s−q), where L_p( (x,y) ) = (|x|^p+|y|^p)^{1/ p}. What happens to the Voronoi diagram of S if p gets close to zero? Surprisingly, although for p close to zero, L_p and L_−p have very different values, they induce almost the same Voronoi diagram. In fact, as p approaches zero from above or from below, the Voronoi diagram converges to the Voronoi diagram of the distance function L_*( (x,y) ) = |xy|. The proof can be found in the following paper:

• The limit of L_p Voronoi diagrams as p→0 is the bounding-box-area Voronoi diagram.
  By Herman Haverkort and Rolf Klein.
  Computing Research Repository (arXiv.org), 2207.07377, 2022.
  text; GeoGebra worksheet

Therefore we propose to define the geometric L₀ distance by L₀( (x,y) ) = |xy|. Thus, we have the following special cases of the (geometric) L_p distance:

L_−∞( (x, y) ) = min( |x|, |y| )
L₀( (x, y) ) = product( |x|, |y| ) (not to be confused with other definitions!)
L₁( (x, y) ) = sum( |x|, |y| ) (Manhattan metric)
L₂( (x, y) ) = √( |x|² + |y|² ) (Euclidean metric)
L_∞( (x, y) ) = max( |x|, |y| )