Distanceeven

Distanceeven is a concept used in discrete mathematics to refer to a distance-related property in which the distance function yields only even integers for all distinct points in a given set. It is not a standard axiom of metric spaces, but a constraint considered in theoretical investigations and toy models.

Formally, a metric space (X, d) is distanceeven if d(x,y) is an even integer for all x ≠ y in X. In particular, the set of distances {d(x,y): x ≠ y} is a subset of the even integers.

One simple way to obtain a distanceeven metric is to start with any metric d on X taking integer values and define d'(x,y) = 2d(x,y). Then d' is a distanceeven metric. Another method is to embed X into Z^k with a distance like Manhattan or Euclidean and constrain the embedding so that all coordinate differences are even; distances then become even integers.

Distanceeven spaces force certain parity constraints. For instance, in spaces with more than two points, triangle inequality and parity together constrain possible configurations. However, not every finite set with a metric can be made distanceeven without rescaling or re-embedding.

Example: Take X = {a,b,c} with d(a,b)=2, d(a,c)=4, d(b,c)=2; all distances are even. Example of construction: X as a subset of Z^2 with the L1 distance under coordinates restricted to even integers.

Applications are primarily theoretical, used to study parity effects in routing, coding theory, or algorithm design. It relates to parity, even graphs, and metric embeddings, and can be contrasted with general metrics where distances may be odd.