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Assume I want to define length of plane curves axiomatically. It seems to be reasonable to assume that

  • The length of a unit segment is 1;
  • Congruent curves have equal lengths;
  • Length is additive with respect to concatenation.

However this is not enuf to define length completely: many different length-like functionals satisfy these properties.

What would be a complete set of axioms?

Motivation. I noticed that many (if not all) proofs of the Crofton formula cheat by assuming implicitly that there is a unique length functional that satisfies the above property, which is wrong. The problem is easy to fix, but the proof I see relies on the constructive definition of length; therefore this extra argument has to be repeated in each variation of the Crofton formula, which is not nice.

P.S. It seems that the following set of axioms solves the problem (thanks to Taras Banakh):

  • The length of any curve is non-negative and invariant with respect to reparametrizations.
  • The length of a unit segment is 1;
  • Congruent curves have equal lengths;
  • Length is additive with respect to concatenation;
  • Length is lower semi-continuous with respect to pointwise convergence.
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  • $\begingroup$ How would you compute the length of a segment of irrational length? You could assume length scales in the correct way, but it seems like some sort of approximation hypotheses would be more natural. $\endgroup$
    – RBega2
    Aug 25, 2018 at 19:05
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    $\begingroup$ What about replacing additivity by $\sigma$-additivity and adding the condition of semicontinuity: If a sequence of curves $(K_n)$ tends to a curve $K_\infty$ in the Vietoris topology, then $length(K_\infty)\le\liminf_{n\to\infty}length(K_n)$? $\endgroup$ Aug 25, 2018 at 19:40
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    $\begingroup$ @AntonPetrunin Here a problem arizes: what is a curve? If it is just a Peano continuum, then the length can be defined as in my answer. But if a curve is a function $\gamma:[a,b]\to \mathbb R^2$, then things became more complicated. For example, what is the length of the curve $\gamma:[-1,1]\to\mathbb R$, $\gamma:t\mapsto |t|$? 1 or 2? What is the definition of a segment? Just an affine function? Then we should add an invariance under the change of a parametrization, etc, etc. $\endgroup$ Aug 25, 2018 at 20:58
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    $\begingroup$ We can define the length of a continuous curve $\gamma$ as the least upper bound of $$\sum_{i=1}^n d(\gamma(x_{i-1}),\gamma(x_i))$$ over all partitions with $$0 = x_0 < x_1 < \cdots < x_{n-1} < x_n = 1.$$ Busemann's Geometry of Geodesics showed that this works in any G-space, so in particular in any Euclidean space. By comparison with this, I'd find any other definition overly complicated. $\endgroup$
    – Matt F.
    Aug 25, 2018 at 21:14
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    $\begingroup$ @alvarezpaiva that is right, but it requires some analysis (say you will need Vitali covering theorem which is nice, but not for a geometry course). $\endgroup$ Sep 4, 2018 at 18:18

1 Answer 1

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I would suggest the following axioms.

The length in a metric space $X$ is a function $\ell:c(X)\to[0,+\infty]$ defined on the family $c(X)$ of all connected compact subsets of $X$ that satisfies the following axioms:

1) $\ell$ is non-degenerated, which means that a continuum $C\in c(X)$ is a singleton if and only if $\ell(C)=0$;

2) $\ell$ is monotone, which means that $\ell(A)\le \ell(B)$ for any continua $A\subset B$ in $X$;

3) $\ell$ is additive, which means that $\ell(A\cup B)=\ell(A)+\ell(B)$ for any continua $A,B\subset X$ with finite non-empty intersection $A\cap B$;

4) $\ell$ is affine, which means that $\ell(f(C))=\lambda\cdot\ell(C)$ for any continuum $C\subset X$, any $\lambda>0$ and any bijective function $f:X\to X$ such that $d(f(x),f(y))=\lambda \cdot d(x,y)$ for all $x,y\in X$;

5) $\ell$ is semicontinuous in the sense that for any $A\in c(X)$ and any $\varepsilon>0$ there a neighborhood $O_A\subset c(X)$ of $A$ in the Vietoris topology such that $\ell(A')\ge \ell(A)-\varepsilon$ for every $A'\in O_A$.

I hope that the following theorem of existence and uniqueness holds:

Theorem. In each Euclidean space $E$ there exists a length $\ell$. Moreover, two lengths $\ell,\lambda:c(E)\to[0,+\infty]$ are equal if $\ell([a,b])=\lambda([a,b])$ for some distinct points $a,b\in E$.

In his survey paper Murat Tuncali writes that the length of continua was studied by Eilenberg, Harrold (1943) and later Buskirk, Nikiel, and Tymchatyn (1992).

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  • $\begingroup$ Originally I wanted to find a right way to fix a gap in the standard proof of Crofton formula --- your suggestion seem to be an overkill. $\endgroup$ Aug 25, 2018 at 21:28

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