TITLE: Why is the Euclidean metric the natural choice?
QUESTION [10 upvotes]: I'm trying to come up with a rationalisation for using the Euclidean distance in an application of mine. Any thoughts on why it is the fundamental choice?
Thanks

REPLY [8 votes]: Beginning with the one-dimensional case: On ${\mathbb R}$ we have translations $t\mapsto t+a$, $\, a$ fixed, and reflexions $t\mapsto -t$ as natural "geometric" isomorphisms. A translation and reflexion invariant metric then necessarily is of the form $d(x,y)=\phi\bigl(|x-y|\bigr)$ where $\phi$ should satisfy some "technical" conditions to make $d$ a metric which in addition is compatible with the inborn topological structure of ${\mathbb R}$. There are many such $\phi$; e.g. the definition $d(x,y):=\tanh\bigl(|x-y|\bigr)$ would turn ${\mathbb R}$ into a bona-fide metric space where all distances are $<1$.
But in ${\mathbb R}$ we have  an additional set of "geometric" isomorphisms, namely scalings. If we want that our metric behaves in a reasonable way under scalings $T_\lambda: \ x\mapsto \lambda x$, $\,\lambda>0$ fixed, then the only $\phi$s left are the functions $\phi(u)= cu$, $\, c>0$ fixed, and we may as well choose $c=1$, so that we arrive at $d(x,y)=|x-y|$.
Now the two-dimensional case: Symmetry considerations like the above imply that we should choose a direction dependent $\phi:\ S^1\to {\mathbb R}_{>0}$ which is even and satisfies a certain convexity condition; then we should put
$$d(x,y):=\phi\left({x-y\over |x-y|}\right)\ |x-y|\ .$$
This metric is translation invariant and behaves correctly under scalings. 
But again, in ${\mathbb R}^2$  new sets of "geometric" isomorphisms are available, namely  compact one-parameter groups of "rotations". If we want our $d$ to be invariant under such a group the only candidates left are of the form
$$\|x\|^2:=\bigl(d(x,0)\bigr)^2= x'Qx\ ,$$
where $Q$ is a positive definite quadratic form of the coordinate variables $x_1$, $x_2$. Introducing a  coordinate system adapted to the $Q$ at hand we then arrive at $\|x\|^2=x_1^2+x_2^2$, i.e., the euclidean distance function.