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 infρF(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
FROSINI, PATRIZIO;
2011-01-01
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 infρF(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.File in questo prodotto:
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