Definition:Euclidean Plus Metric

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

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

The Euclidean plus metric $d: \R \times \R \to \R$ is the metric defined as:

$\map d {x, y} := \set {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$

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

Also see

Linguistic Note

The Euclidean plus metric appears not to be allocated a specific name in the literature.

The name Euclidean plus metric has been coined for $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to allow it to be referred to compactly.