Intrinsic Quasi-Metrics

Abstract

The point pair function $$p_G$$ p G defined in a domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n is shown to be a quasi-metric, and its other properties are studied. For a convex domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n , a new intrinsic quasi-metric called the function $$w_G$$ w G is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.

Intrinsic metrics in ring domains

“…Acknowledgements This research continues my earlier work in [13][14][15][16][17]. Three last mentioned of these articles have been co-written with Professor Matti Vuorinen, to whom I am indebted for all guidance and support.…”

“…Thus, in order to study the intrinsic geometry of the ring R(r , 1), we use here a few known generalizations of the B Oona Rainio ormrai@utu.fi 1 Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland hyperbolic metric: the j * -metric found first in [7] by modifying the distance ratio metric introduced in 1979 by Gehring and Palka [5], the triangular ratio metric introduced by Hästö in 2002 [9] and recently studied in [1,[14][15][16][17], and the Möbius metric originally introduced in [20, pp. 115-116] and extensively studied by P. Seittenranta in his PhD thesis [18].…”

Introducing a New Intrinsic Metric

“…More potential properties for a hyperbolic type metric are presented in [7, p. 191-192]. Examples of a hyperbolic type metric include the triangular ratio metric studied in [12][13][14], the Barrlund metric [4] and the Cassinian metric [10], whereas this work focuses on the following new intrinsic metric.…”

Möbius metric in sector domains

“…It means that, given two points in a domain, we do not only consider how close these points are to each other but also how they are located with respect to the boundary of the domain. In order to measure these kinds of distances, we need to use suitable intrinsic or hyperbolic type metrics, which have been recently studied, for instance, in [1,5,6,7,12,13,14].…”