I'm trying to understand better the following sentence (see link to the reference at the end):

To represent the metric of $n$ points in the Euclidean space, one clearly needs no more than $n−1$ dimensions.

1. What "representing a metric" means?
2. If (1) means to store the "distance" from a point to the origin (or the point "length", like $l_2$), how is it possible to do it with $n-1$ dimensions (instead of $n$)?

Reference: "Dimensionality Reductions in $l_2$ that Preserve Volumes and Distance to Affine Spaces", by Avner Magen, section 1 ("Introduction").

I'm trying to understand better the following sentence (see link to the reference at the end):

To represent the metric of $n$ points in the Euclidean space, one clearly needs no more than $n−1$ dimensions.

What "representing a metric" means?

Reference: "Dimensionality Reductions in $l_2$ that Preserve Volumes and Distance to Affine Spaces", by Avner Magen, section 1 ("Introduction").