Revisiting separation properties of convex fuzzy sets

摘要

Separation of convex sets by hyperplanes has been extensively studied on crisp sets. In a seminal paper from L. A. Zadeh [1] separability and convexity are investigated, however there is a flaw on the definition of degree of separation. We revisited separation on convex fuzzy sets that have level-wise (crisp) disjointness with non-empty interior at certain level and introduced the concept of minimal level of separation for such fuzzy sets. On this context, the smallest level in which a separation by a hyperplane occurs coincides with the maximal degree of the (fuzzy) intersection. Moreover, this property suggests an algorithm for finding the maximal grade of a (fuzzy) intersection based on hyperplane separability level-wise of fuzzy sets.