Let $U\subset\mathbb{R}^n$ be a bounded, open and path-connected set. Then every two $x,y\in U$ can be connected to each other by a polygonal chain $l$. I am wondering whether you could find an $M>0$ such that every $x,y\in U$ can be connected to each other by such a polygonal chain that the length of $l$ satisfies $|l| < M|x - y|$? That is, that some family of paths connecting points of $U$ are somewhat proportional to the metric distance between the points $x$ and $y$.
I haven't made any progress with this problem since I am not sure what would be such a uniform bounding constant for all two pairs of points $x, y\in U$.
That is (trivially) true if $n=1$ since the connected sets in $\Bbb R$ are exactly the intervals, but it is false if $n \ge 2$.
For $n = 2$ let $U$ be the disk of radius $2$ with the negative real axis removed, and $x, y$ be the points $(-1, \pm \epsilon)$ with some $0 < \epsilon < 1$. Their distance is $2 \epsilon$, but every path connecting these points in $U$ must “go around the origin” and has a length of at least $2$.
In higher dimensions one can use the same idea and remove half of a hyperplane from a $n$-dimensional disk.