Some geometric properties and exhaustion of John disk (Q2785283)

A simply connected bounded domain \(D\) in the plane is a \(c\)-John disk if there exist a point \(z_o\in D\) and a constant \(c\geq 1\) such that each point \(z_1\in D\) can be joined to \(z_o\) by an arc \(\gamma\) in \(D\) satisfying \(l(\gamma(z_1,z))\leq c \text{ dist}(z,\partial D)\) for each \(z\in \gamma\), where \(l(\gamma(z_1,z))\) is the length of the subarc of \(z\in \gamma\) with endpoints \(z_1,z\). The author presents some geometric properties of John disks.