Characterizations of John spaces (Q667562)

John domains, see [\textit{F. John}, Commun. Pure Appl. Math. 14, 391--413 (1961; Zbl 0102.17404)] and [\textit{O. Martio} and \textit{J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I, Math. 4, 383--401 (1979; Zbl 0406.30013)], or their generalizations, John spaces, are studied in the setup of a general metric space \((X, d)\). It is assumed that \(X\) is a noncomplete, rectifiably connected and locally \((\lambda, c)\)-quasiconvex metric space. The last condition means that there are \(\lambda \in (0, 1/2]\) and \(c \geq 1\) such that for all \(x \in X\) each pair of points in \(B(x,\lambda d(x))\) can be joined by a \(c\)-quasiconvex path. Here \(d(x) = d(x, \partial X)\) and \(\partial X\) consists of the completion of \(d\) to the noncomplete points. Now \(X\) is called a length \(a\)-John space with center \(x_0 \in X\) if for every \(x \in X\) there is an \(a\)-carrot arc, \(a \geq 1\), joining \(x\) to \(x_0\). An \(a\)-carrot arc is a rectifiable arc \(\alpha\) such that for all \(z \in \alpha\), \(l(\alpha[x,z]) \leq a \, d(z)\). In the paper [\textit{F. W. Gehring} and \textit{K. Hag}, Complex Variables, Theory Appl. 9, No. 1--3, 175--188 (1987; Zbl 0638.30022)], equivalent definitions for a length \(a\)-John space are given all based on metric geometry. In particular, there is a condition which guarantees that the so called diameter carrots where \(l(\alpha[x,z])\) is replaced by \(\operatorname{diam}(\alpha[x,z])\) leads to the same result as the length \(a\)-carrot arcs, i.e., it is shown that if for every \(x \in X\) one can join \(x\) to the distinguished point \(x_0\) by a diameter carrot where the curve \(\alpha\) satisfies the \(\varphi\) natural condition, \(\operatorname{diam}_k(\gamma[x,y]) \leq \varphi(\operatorname{diam}\operatorname{dist}( \alpha[x,y])/ \operatorname{dist}(\gamma[x,y], \partial X)\), then \(X\) is a length \(a\)-John space. The function \(\varphi: [0, \infty) \rightarrow [0, \infty)\) is an increasing function and \(\operatorname{diam}_k\) refers to the quasihyperbolic metric in \(X\), see [\textit{F. W. Gehring} and \textit{B. P. Palka}, J. Anal. Math. 30, 172--199 (1976; Zbl 0349.30019)]. It is also shown that if \(f: X \rightarrow X'\) is an \(\eta\)-quasisymmetric homeomorphism and \(X\) and \(X'\) satisfy the above conditions and if \(X\) is a length \(a\)-John space, then \(X'\) is a length \(b\)-John space with \(x_{0}' = f(x_0)\) and \(b\) depends on \(\lambda\), \(c\), \(\eta\) and \(a\).