On the nonexistence of Hausdorff-like metrics for fuzzy sets

Authors:
Peter Brass
Affiliations:
Fachbereich Mathematik and Informatik, Institut für Informatik, Freie Universität Berlin, Takustrasse 9, D-14195 Berlin, Germany
Venue:
Pattern Recognition Letters
Year:
2002

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Abstract

There has been a number of papers proposing different extensions of the Hausdorff metric to fuzzy sets. None of these proposals behave as one would intuitively expect. It is the aim of this paper to start a systematic study of the possible behaviour of metrics on fuzzy sets, and to show that under a reasonable set of axioms, like the metric space axioms and additionally invariance under motions and independence of the length unit, there is no metric on the fuzzy subsets of a Euclidean space.