##
Voronoi Diagram

##
with the Minkowski Metric L_{p}

The Minkowski (power) metric from a point
p to a point p_{i} in
R^{m} is defined
by

dL_{p}(p, p_{i}) = [_{j=1}∑^{m}
|* x*_{j} - *x*_{ij} |^{p}]^{1/p}
--------------- 1

where (*x*_{1}, *x*_{2}, . . . ,
*x*_{m}) and (*x*_{i1}, *x*_{i2}, . . . ,
*x*_{im}) are the Cartesian coordinates of p and p_{i},
respectively. Customarily the symbol L_{p} is used
for the Minkowski metric, where
p refers to the degree of the power.

The parameter p varies in the range of
1 ≤ p < ∞.

- If p = 1, equation 1 becomes

dL_{1}(p, p_{i}) = _{j=1}∑^{m}
|* x*_{j} - *x*_{ij} |
--------------- 2

which is called the
Manhattan metric, the city-block distance or the
taxi-cab distance.

- If p = 2,
equation 1 becomes

dL_{2}(p, p_{i}) = [_{j=1}∑^{m}
|* x*_{j} - *x*_{ij} |^{2}]^{1/2}

which is called the Euclidean distance.

- If p = ∞, the equation 1 becomes

dL_{∞}(p, p_{i}) = [max_{j}
{
|* x*_{j} - *x*_{ij} | j
I_{m} }

which is called the Supermum metric or
dominance metric.