Sendograph metric and relatively compact sets of fuzzy sets

Let 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” (called sendographs). Fan [Fuzzy Sets and Systems 143 (2004) 471–477] characterized compact sets of fuzzy numbers (=convex fuzzy subsets of the real line R) with respect to sendograph metric and declared open the problem of finding a criterion for convex fuzzy subsets of Rn. In this paper relatively compact subsets of (K(Y),H) are characterized (see Theorem 8) and, consequently, it is shown that Fan's theorem holds for arbitrary metric spaces without any convexity assumption. In the proofs a variational convergence (called Γ-convergence), introduced by De Giorgi–Franzoni [Atti. Accad. Naz. Lincei Rend. Cl. Sc. Mat. Fis. Natur. 58(8) (1975) 842–850; Rend. Sem. Mat. Brescia 3 (1979) 63–101], and author's previous results are fundamental; no isometrical embedding theorem is used.