A generalization of the Minkowski embedding theorem and applications

Purl and Ralescu (1985) gave, recently, an extension of the Minkowski Embedding Theorem for the class E-L(n) of fuzzy sets u on R-n with the level application alpha --> L(alpha)u Lipschitzian on the C([0, 1] x Sn-1) space. In this work we extend the above result to the class E-C(n) of level-continuous applications. Moreover, we prove that E-C(n) is a complete metric space with E-L(n) not subset of or equal to E-C(n) and <(E-L(n))over bar> = E-C(n). To prove the last result, we use the multivalued Bernstein polynomials and the Vitali's approximation theorem for multifunction. Also, we deduce some properties in the setting of fuzzy random variable (multivalued). (C) 1999 Elsevier Science B.V. All rights reserved.