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I have the following question: for a general domain $\Omega$ in $\mathbb{R}^n$, is it true that for each pair of points $x,y\in \Omega$, there exists a curve $\gamma$ connecting $x$ and $y$ in $\Omega$, almost length minimizing in the sense that, for each pair $z,w\in \gamma$, the length of $\gamma_{zw}$ satisfies $$\text{length}(\gamma_{zw})\leq 2\inf_{\beta \text{ connects }z, w \text{ in } \Omega}\text{length}(\beta).$$ Here $\gamma_{zw}$ denotes the restriction of $\gamma$ from $z$ to $w$.

Thanks for any comments and suggestions.

Edit: the topic is closed.

Here one needs piecewise linear approximation of a general curve.

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  • $\begingroup$ Are you requiring $\gamma$ to be simple? If not, then what does $\gamma_{z w}$ mean? $\endgroup$
    – LSpice
    Commented Dec 5, 2020 at 2:03
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    $\begingroup$ Yes, you can fix a $\gamma$ that almost minimizes the total length, and then approximate by straight line segments in small balls (cover $\gamma$ by finitely many of these). $\endgroup$
    – Christian Remling
    Commented Dec 5, 2020 at 2:21
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    $\begingroup$ Please delete the question; here it's just self-vandalized. I rolled back to original question and added your sentence. $\endgroup$
    – YCor
    Commented Dec 5, 2020 at 7:51
  • $\begingroup$ I disagree with the closure of this question. The formulation needs to be slightly improved BUT the answer is not trivial as what the comments suggest. I will post a proof sooner or later. $\endgroup$
    – Romain Gicquaud
    Commented Dec 8, 2020 at 10:53

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