## The context

I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following.

Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its length metric.¹

If the distortion $\sup_{x\neq y}\frac{d_\ell(x,y)}{d_{\mathbb R^d}(x,y)}$ is less than $\frac\pi2$, then $K$ is simply connected.

The proof, if I understand correctly, goes roughly as follows [Edit: see below for more detail]:

- suppose there exists a compact $K$ satisfying the inequality but admitting a non-trivial loop;
- choose a non-trivial loop $\gamma$ of minimal length;
- show that the length metric on $\gamma$ is the same as that of $K$ (by minimality of the loop);
- argue that $\gamma$ must ‘make a turn’ at some point, contradicting the metric assumption.

## My question

lies in Point 2 of this program. Of course, there is no reason for such a curve to exist (see e.g. the Hawaiian earring). [Edit: YCor's example, in the comments, shows in fact that even in a fixed free homotopy class, one may not find a length-minimising curve.]

Am I mistaken in thinking that Point 2 is actually what is happening in the proof?

[Edit: This point is somewhat answered by M. Dus in his answer, who notes that it might be suggested by the authors that the subset is supposed to be a submanifold, or a domain with smooth boundary.]

If not, is there a simple way to get the argument back on track?

[Edit: Following M. Dus's answer, the question might actually be: Is there a simple way to get the argument working for all compact sets?]

## A bit more about the proof

I'll start by saying that a quick search in your favourite search engine will take you to a pdf version of the book, for anyone interested. However I thought I should make my question as self-contained as possible.

Point 1 of the proof is self-evident, and point 2 is given as is in the book (‘let $\alpha$ be a nontrivial homotopy class in which there exists a curve of minimal length among all homotopically nontrivial loops’). Say that we get a curve $\gamma$ of length $\ell$, parametrised by arc length by $c:\mathbb R/\ell\mathbb Z\to\mathbb R^d$.

Point 3 (after trimming) aims to show that the path length distance between $c(t)$ and $c(t+\ell/2)$ is the same, viewed as points in $K$ or $\gamma$. Otherwise, there must be a curve in $K$ of length $<\ell/2$ between these two points, creating two loops of length $<\ell$ whose product is $c$. One of them must be non trivial (because $c$ is), a contradiction (because $c$ is of minimal length among such curves).

Point 4, then, defines $\tilde c:t\mapsto c(t+\ell/2)-c(t)$, $r=|\tilde c|$ and $u=\tilde c/|\tilde c|$. We will show that the length of $u$ is less than $2\pi$, a contradiction, because $u$ takes values in the sphere, and $u(t+\ell/2)=-u(t)$.

Noting that $\frac{\mathrm du}{\mathrm dt}$ is orthogonal to $\tilde c$, we get $$ \left|\frac{\mathrm du}{\mathrm dt}\right|^2 = \left|\frac1{r(t)}\frac{\mathrm d\tilde c'}{\mathrm dt}\right|^2 - \left|\frac{-r'(t)}{r(t)^2}\tilde c(t)\right|^2 \leq \frac{4-r'(t)^2}{r(t)^2} \leq \frac4{r(t)^2}. $$ But according to Point 3, the path length distance in $K$ between $c(t)$ and $c(t+\ell/2)$ is precisely $\ell/2$, so we get $$ \frac{\ell/2}{r(t)}\leq\sup_{x\neq y}\frac{d_\ell(x,y)}{d_{\mathbb R^d}(x,y)}\leq\frac\pi2-\varepsilon<\frac\pi2. $$ This yields $|u'(t)|\leq(2\pi-\varepsilon)/\ell$ so the length of $u$ is at most $2\pi-\varepsilon<2\pi$, as announced.

¹ If $S\subset\mathbb R^d$, its associated length metric is $$d_\ell(x,y):=\inf\{\mathrm{length}(\gamma),\gamma\text{ a path from }x\text{ to }y\text{ with values in }S\}.$$ For instance, $d_\ell$ is infinite if $x$ and $y$ are not in the same connected component, or two different points in the Koch snowflake.

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