I'm looking for a quick, snap-your-fingers proof of the following result:

A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm.

To be clear about the terminology: I'll say that a metric $d$ comes from a norm if $d(x,y) = \|x - y \|$.

I'm also looking for a reference where the following question is addressed:

Given a continuous bi-invariant length metric on a Lie group, is it true that one-parameter subgroups are geodesics and/or that the metric comes from a bi-invariant continuous Finsler metric?

**Remark 1.** Please pay attention to the level of generality. These things are really easy if you assume a bit more regularity than one should.

**Edit: May 12, 2014.** I had written earlier that I had a proof which used
convolution and the action of the affine or linear group on translationn-invariant distances to regularize the metric. In fact, *this approach is wrong or incomplete.* It is not clear (or not true?) that the regularized metric is a length metric.

On the other hand, the OP does follow from the much, more general results of Berestovskii cited by Yves in his comment.

left-invariant length metrics on connected Lie groups. Link: link.springer.com/article/10.1007%2FBF00972413 $\endgroup$8more comments