Definition:Euclidean Plus Metric

Definition
Let $\R$ be the set of real numbers.

Let $\left\{{r_i}\right\}$ be an enumeration of the rational numbers $\Q$.

Let $d: \R \times \R \to \R$ be the metric defined as:


 * $d \left({x, y}\right) := \left|{x - y}\right| + \displaystyle \sum_{i \mathop = 1}^\infty 2^\left({-i}\right) \inf \left({1, \left|{ \max_{j \mathop \le i} \frac 1 {\left|{x - r_j}\right|} - \max_{j \mathop \le i} \frac 1 {\left|{y - r_j}\right|} }\right| }\right)$

Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\Q$.

Linguistic Note
This metric is not named in the literature. The name Euclidean plus metric has been coined for in order to allow it to be referred to compactly.

Also see

 * Euclidean Plus Metric is Metric