Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$


We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a curve $\gamma$ in $D$ satisfying $\text{length}(\gamma) \le C|x-y|.$ Here, $\text{length}(\gamma)$ denotes the length of $\gamma$.


The property P appears in the Whitney extension theorem (see Theorem in [W](https://www.jstor.org/stable/pdf/1968745.pdf?refreqid=excelsior%3Ab0aedd84cb53f3564899ca902fad3d0a&ab_segments=&origin=&initiator=&acceptTC=1) for details), and is sometimes referred to the quasi-convexity.


It is known that bounded Lipschitz domains are quasi-convex.


Can we show that unbounded Lipschitz domains are also quasi-convex?



On the property P in the Whitney extension theorem
No. The complement $M$ in $\mathbb R^2$ of the union of the closed unit ball and the *half strip* $\{(x,y)\in\mathbb R^2: x\ge 0, |y|\le 1\}$ has $C^1$-boundary but curves in $M$ joining $(n,2)$ and $(n,-2)$ (which have distance $4$) have length $>2n$.


