Sendograph metric and relatively compact sets of fuzzy sets (Q812617)

Let \(\mathbb{K}(Y)\) be the family of all fuzzy subsets of an arbitrary metric space \(Y\), which are upper-semicontinuous, normal with nonempty compact support. The sendograph distance \(H\) between fuzzy sets is the Hausdorff distance of their ``reduced hypographs''. \textit{T. Fan} [Fuzzy Sets Syst. 143, 471--477 (2004; Zbl 1044.03042)] characterized compact sets of fuzzy numbers with respect to the sendograph metric and declared open the problem of finding a criterion for convex fuzzy subsets of \(\mathbb{R}^n\). In this paper, relatively compact subsets of \((\mathbb{K}(Y),H)\) are characterized and it is shown that Fan's theorem holds for arbitrary metric spaces without any convexity assumption.