Definition:Euclidean Plus Metric

From ProofWiki
Jump to navigation Jump to search


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

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

The Euclidean plus metric $d: \R \times \R \to \R$ is 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 $\Q$.

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.