I was again thinking about the famous $\pi=4$ paradox, and this question in particular: How to convince a layperson that the $\pi = 4$ proof is wrong?, about why the standard sup over polygonal approximation definition is "more correct" than any other polygonal approximation definition (e.g. the one that leads to $\pi=4$). A couple of comments in that answer caught my eye, namely one from @MarcelBesixdouze: "Euclidean distance we talk about in every day life: the one that is invariant under rigid rotations/translations", and some others mentioning scale invariance.
So my question is: is there a characterization of the standard length function $\ell : C^0([0,1],\mathbb R^n) \to [0,\infty]$ defined as $\ell(\gamma) := \sup_{0=t_0 < \ldots < t_n=1} \sum_{i=1}^n |\gamma(t_j)-\gamma(t_{j-1})|$ (sidenote: this definition can be extended to any metric space $X$ in place of $\mathbb R^n$, I think?), as say the unique function $\ell: C^0([0,1],\mathbb R^n) \to [0,\infty]$ satisfying
- invariance under rigid motion (i.e. for any rigid motion $T$ on $\mathbb R^n$, $\ell(T(\gamma)) = \ell(\gamma)$)
- independent of parameterization of $\gamma$
- "commutativity with sums" $\ell(\gamma_1+\gamma_2) = \ell(\gamma_1)+\ell(\gamma_2)$ (rigorous definitions of parameterization, sums, etc. can be found here)
- "agrees with lengths for lines" for any line segment $L$ between two points $x,y$, $\ell(L) = |x-y|$
- more/fewer conditions?
If there is such a characterization, these properties intuitively feel correct, and therefore would be "sufficient evidence" to support that the standard notion of arclength is indeed "the correct notion", as opposed to any other competing definition.
EDIT: if we further assume the "lines are the shortest path between two points" axiom, i.e. $\ell(\gamma) \geq |\gamma(1)-\gamma(0)|$, then the addition axiom would lead to $\ell(\gamma) \geq \ell(PL)$ where $PL$ is any piecewise linear interpolation of $\gamma$. Then the standard length function is the minimal such length function, by definition of sup. But then I'm not so confident that our assumption about lines being the shortest paths can be fully justified...related: Why is a straight line the shortest distance between two points?. That observation can be proven rigorously using the triangle inequality for polygonal paths, but for arbitrary paths, it must be assumed. Then again perhaps I shouldn't be too taken aback, as all (most?) infinite objects in mathematics are really just limits of finite things.
EDIT 3/22/22: although it is NOT true that the standard definition of the arclength of a curve $\gamma$ is the limit of the arclengths of polygonal approximations to $\gamma$ (limit in terms of polygonal paths getting closer and closer in the sup norm), since one can define polygonal paths of arbitrarily long arclengths but remain arbitrarily close to $\gamma$ ("walking-the-dog paths"), there still is an interesting pattern that emerges (I have not proven the following results; merely intuited them):
for a given $\epsilon>0$, the set of all arclengths of polygonal approximations within $\epsilon$ of $\gamma$ form an interval like $(l_\epsilon,\infty)$, where in fact the lower bounds $l_\epsilon$ increase (perhaps not strictly) for sufficiently small $\epsilon \searrow 0$. This limit of the lower bounds I think is exactly the standard arclength of $\gamma$.
In other words, we exploit the natural asymmetry of the situation (the fact that the arclengths of the polygonal approximations within $\epsilon$ are bounded below but not bounded above) to give credence to the idea that our definition of arclength is somehow "canonical", or "natural".
