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.