A New Intrinsic Metric on Metric Spaces

A new intrinsic metric called S D -metric is introduced for a general metric space ( X , d ), where D is a non-trivial bounded closed subset of X . The metric S D can be used to define a strongly hyperbolic metric on X . We consider the convergence of metric spaces { ( X , S D n ) } n = 1 ∞ for a sequence of non-trivial bounded closed subsets { D n } n = 1 ∞ . The distortion property of the new metric on the unit ball B n is also studied under the Möbius transformations of the unit ball.