A generalization of the Minkowski embedding theorem and applications

Purl and Ralescu (1985) gave, recently, an extension of the Minkowski Embed ding Theorem for the class E-L(n) of fuzzy sets u on R-n with the level app lication 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-continuo us 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 a nd the Vitali's approximation theorem for multifunction. Also, we deduce so me properties in the setting of fuzzy random variable (multivalued). (C) 19 99 Elsevier Science B.V. All rights reserved.