John domains and the John constant

Assume Countable Choice. Let $n\ge 2$ and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, bounded and nonempty. Its boundary $\mathrm{\partial }\mathrm{\Omega }$ is then a nonempty compact subset of ${\mathbb{R}}^n$ contained in a sufficiently large closed ball (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact), and the distance $\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })$ is defined for every $x$ and is a $1$-Lipschitz function of $x$ ($\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz).

Rectifiable curves, their arclength functions are used with the conventions of The arc-length function $s_{\gamma }(t)=L(\gamma {\mathrm{\mid }}_{[a,t]})$ of a rectifiable path.

John domain and John constant. A pair $(\mathrm{\Omega },x_0)$ with $x_0\in \mathrm{\Omega }$ satisfies the John condition with constant $c\ge 1$ if for every $x\in \mathrm{\Omega }$ there is a rectifiable curve $\gamma :[0,l]\to \mathrm{\Omega }$ parametrised by arclength, with $\gamma (0)=x$, $\gamma (l)=x_0$ and $\mathrm{dist}(\gamma (t),\mathrm{\partial }\mathrm{\Omega })\ge c^{-1}\mathrm{\mid }x-\gamma (t)\mathrm{\mid }\text{for every }t\in [0,l]\mathrm{.}$ Write $c_J(\mathrm{\Omega },x_0)$ for the infimum of the admissible constants $c\ge 1$; with the convention $\inf \mathrm{\varnothing }=+\mathrm{\infty }$, this value belongs to $[1,+\mathrm{\infty }]$. The domain $\mathrm{\Omega }$ is a John domain if $c_J(\mathrm{\Omega },x_0)<\mathrm{\infty }$ for some $x_0\in \mathrm{\Omega }$, and $c_J(\mathrm{\Omega },x_0)$ is the John constant of the pair $(\mathrm{\Omega },x_0)$. The estimates below use an admissible constant, never the finiteness of the infimum alone, and whether the infimum is attained is immaterial.

Arclength form. If $\gamma$ is arclength parametrised from $x$, then $t\ge \mathrm{\mid }\gamma (t)-x\mathrm{\mid }$ for $0\le t\le l$, because the straight segment from $x$ to $\gamma (t)$ is no longer than the curve. Hence a curve satisfying the arclength normalisation $\mathrm{dist}(\gamma (t),\mathrm{\partial }\mathrm{\Omega })\ge c^{-1}t$ for all $t$ also satisfies the displayed relative-distance condition with the same constant $c$. Only this implication is used here; the chain construction below uses the displayed relative-distance condition.

Connectedness. A John domain is path-connected and hence connected: given $x,y\in \mathrm{\Omega }$, choose curves ${\gamma }_x$ from $x$ to $x_0$ and ${\gamma }_y$ from $y$ to $x_0$ as in the definition (under Countable Choice the two curves may be chosen simultaneously) and traverse ${\gamma }_x$ followed by the reverse of ${\gamma }_y$. No regularity of $\mathrm{\partial }\mathrm{\Omega }$ is assumed beyond what the definition uses.