Location: Arkansas Children's Nutrition CenterTitle: Revisiting separation properties of convex fuzzy sets
|GOMEZ-ACEVEDO, HORACIO - Arkansas Children'S Nutrition Research Center (ACNC)|
Submitted to: Journal of Intelligent & Fuzzy Systems
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 9/1/2015
Publication Date: 10/5/2015
Citation: Gomez-Acevedo, H. 2015. Revisiting separation properties of convex fuzzy sets. Journal of Intelligent & Fuzzy Systems. 29(2):845-849. doi: 10.3233/IFS-151613.
Interpretive Summary: This article corrects a statement on a highly cited paper that claims that two (convex fuzzy) sets can be divided by a plane. We gave a mathematical proof that shows that under some conditions this separation can occur but not under the conditions mentioned on the other paper.
Technical Abstract: Separation of convex sets by hyperplanes has been extensively studied on crisp sets. In a seminal paper 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.