On polygonal fuzzy sets and numbers

In this paper, we propose a new mathematical formalization of the concept of polygonal fuzzy numbers and an extension of this notion to fuzzy sets on ^Rn. We study the mathematical structure of these families of fuzzy sets and show that each family is a complete and separable metric space when endowed with the generalized Hausdorff metric. Moreover, we show that for n=1, the families of polygonal fuzzy numbers are isomorphic to some convex and closed convex cone of a finite dimensional space. We obtain generalizations and extensions of some previous results on polygonal fuzzy numbers and simplified proofs of some well-known results about approximation of fuzzy n-dimensional quantities. Finally, some developments about the approximation of families of fuzzy sets are introduced.