The Hausdorff distance, the Gromov–Hausdorff, the Fréchet and the natural pseudo-distance are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as inf F(ρ) where F is a suitable functional and ρ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space K, in such a way that the composition in K (extending the composition of homeomorphisms) passes to the limit and, at the same time, K is compact.
No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit / P., Frosini; Landi, Claudia. - In: APPLIED MATHEMATICS LETTERS. - ISSN 0893-9659. - STAMPA. - 24:(2011), pp. 1654-1657. [10.1016/j.aml.2011.04.007]
No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit
LANDI, Claudia
2011
Abstract
The Hausdorff distance, the Gromov–Hausdorff, the Fréchet and the natural pseudo-distance are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as inf F(ρ) where F is a suitable functional and ρ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space K, in such a way that the composition in K (extending the composition of homeomorphisms) passes to the limit and, at the same time, K is compact.
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