Intrinsic metric in spaces of compact subsets with the Hausdorff metric
- Irmina Herburt
We prove that, if a metric space (X,ρ) can be endowed with the intrinsic metric ρ∗ (the intrinsic distance of two points is defined as the infimum of the lengths of arcs joining these points), then the Hausdorff metric ρH in the space C(X) of compact subsets of X induces the intrinsic metric (ρH)∗, and the equality (ρH)∗ = (ρ∗)H is satisfied. This implies that ρH = (ρH)∗ if and only if ρ∗ = ρ and that each isometry between spaces X1 and X2 with intrinsic metrics induces an isometry between C(X1) and C(X2) with intrinsic metrics.
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- Kotus Janina Janina Kotus (eds.): 20 Years of the Faculty of Mathematics and Information Science. A collection of research papers in mathematics, 2020, Oficyna Wydawnicza Politechniki Warszawskiej, 194 p., ISBN 978-83-8156-156-3
- Keywords in English
- intrinsic metric, intrinsic isometry, hyperspace of compact sets, Hausdorff metric, arcs in hyperspace of compact sets
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- eng (en) English
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- = 20.0, 18-01-2021, MonographChapterAuthor
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