User:Viennot/Inframetric

In mathematics, an inframetric is a distance function between elements of a set that generalizes the notion of metric. It is defined by the following weaker version of the triangle inequality: d(x, z) &le; $$\rho$$ max{d(x, y), d(y, z)} for some parameter $$\rho$$ &ge; 1. A set with an inframetric is called an inframetric space. This notion subsumes both standard metric spaces (1 &le; $$\rho$$ &le; 2) and ultrametric spaces ($$\rho$$ = 1). Inframetrics were notably introduced to model internet round-trip delay times.

Definition
For a given parameter $$\rho$$ &ge; 1, a $$\rho$$-inframetric on a set X is a function (called the distance function or simply distance)

d : X × X → R

(where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions:


 * 1) d(x, y) ≥ 0     (non-negativity)
 * 2) d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
 * 3) d(x, y) = d(y, x)      (symmetry)
 * 4) d(x, z) ≤ $$\rho$$ max{d(x, y), d(y, z)}      ($$\rho$$-inframetric inequality).

Note that only the last axiom differs from the metric definition. The classical triangle inequality d(x, z) ≤ d(x, y) + d(y, z) implies d(x, z) ≤ 2 max{d(x, y), d(y, z). Any metric is thus a 2-inframetric. The definition of 1-inframetric is equivalent to that of ultrametric.