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Let $V=\mathbb{R}^n$.
Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$.
Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = |\lambda|d(x,y)$.
Is $d$ necessarily induced by a norm?

Motivation: I've been thinking of $\pi$ and thought about why the ratio between a circles's circumference and its radius is constant. The proof is easy and is applicable to any norm. I think the "positive homogeneity" condition I posed on the metric above is enough for this ratio to be constant.

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You did note that this ratio is normally $2\pi$, not? Doesn't render the question invalid, just sayin'. –  Lord_Farin Oct 20 '12 at 15:29
    
I did note that, thanks. I thought the formulation is clearer this way. –  Gils Oct 20 '12 at 15:31
    
This might fail for a non translation invariant metric (i.e. $d(x+c, y+c) \neq d(x,y)$. Have you checked that? –  filmor Oct 20 '12 at 15:40
    
Ah, answered just now by Lord Farin … ;) –  filmor Oct 20 '12 at 15:40

1 Answer 1

up vote 4 down vote accepted

The answer is no. You need translational invariance as well; then it's a pretty well-known theorem (see e.g. here).

As a counterexample when leaving out the translational invariance, consider:

$$d: \Bbb R^n \times \Bbb R^n \to \Bbb R_{\ge 0}: d (x,y)=\begin{cases} \|x\|+\|y\| & \text{if $x \ne y$}\\ 0 & \text{otherwise.} \end{cases}$$

This metric is sometimes referred to as the "metric of the French railway system", although there are similar metrics with the same name (cf. the comments).

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but some of our trains do have intermediate stops ! –  mercio Oct 20 '12 at 15:53
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I suppose it could also be called the "metric of the French intercity service system locally around Paris", but that's such a mouthful. –  Lord_Farin Oct 20 '12 at 15:55
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+1. I thought the SNCF distance $d$ was defined by $d(x,y)=\|x-y\|$ when $x$ and $y$ are colinear, and $d(x,y)=\|x\|+\|y\|$ otherwise. The intermediate stops again... –  Did Oct 20 '12 at 16:13
    
@did: Apparently multiple versions exist; they both work, whatever their name. –  Lord_Farin Oct 20 '12 at 16:15
    
@did’s version is also called the Paris metric. (I usually call it the hedgehog metric.) –  Brian M. Scott Oct 20 '12 at 16:26

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