The authors study a generalization of a metric introduced by P. A. Hästö [J. Math. Anal. Appl. 274, No. 1, 38–58 (2002; Zbl 1019.54011)]. The main result is the following theorem:
Theorem 1.1. Let \(D\) be a nonempty set in a metric space \((X, \rho)\) with nonempty boundary \(\partial D\). Then the function \[ h _{D,c}(x,y) = \log \left( 1+ c \frac{\rho(x,y)}{\sqrt{d_D(x) d_D(y)}} \right), \] where \(d_D(x) = \mathrm{dist}(x, \partial D)\) is a metric for every \(c \geq 0\). The constant \(2\) is best possible.
When \(X\) is the upper half-space \(\mathbb{H}^n\) the metric \(h_{\mathbb{H}^n, c}\) is closely related to the hyperbolic metric.
The proof of the main theorem boils down to verification of the triangle inequality for \( h _{D,c}\).
Some applications of the \(h _{D,c}\)-metrics to quasiconformal homeomorphisms are considered in the last section of the paper.