| Abstract |
We consider the problem of extension of pairs of continuous and bounded, partial metrics
which agree on the non-empty intersections of their domains which are closed and bounded
subsets of an arbitrary but fixed metric space. Two pairs of such metrics are close if their
corresponding graphs are close and if the intersections of their domains are close in the Hausdorff
metric. If, besides, these metrics are uniformly continuous on the intersections of their domains
then there is a continuous positive homogeneous operator extending each such a pair of partial
metrics to a continuous metric on the union of their domains. We prove that, in general, there
is no subadditive extension operator (continuous or not) for such pairs of metrics. We provide
examples showing to what extent our results are sharp and we obtain analogous results for
ultrametrics. |
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