Stabilizing Metrics Induced by Logarithmic Oscillation in Metric Spaces and Gromov Hyperbolicity

We generalize from Euclidean spaces to an arbitrary metric space (X,d), two stabilizing distance functions introduced and studied by Boskoff and Suceavă induced by a version of Barbilian’s logarithmic oscillation. These distance functions are defined on X∖M, where M⊂X is non-empty and M≠X, using an influence function F:X∖M→(0,+∞) as a counterpart of the distance to M and a function f:[0,+∞)→[0,+∞), which vanishes only at the origin, is non-decreasing and subadditive. Under natural conditions on F and f, the stabilizing distance functions studied here are shown to be genuine metrics possessing useful geometric properties, including completeness, controlled quasiconformal distortion and Gromov hyperbolicity. A central result regarding the first stabilizing metric provides an improved hyperbolicity constant compared with the previously known bound in the Euclidean case, under the assumptions that the underlying metric space is Ptolemaic, M is a singleton, F is the distance to M and f is the identity map. Explicit estimates for Gromov hyperbolicity constants of both stabilizing metrics and of a new generalization of Vuorinen’s distance ratio metric are also obtained, in arbitrary metric spaces.