A generalization of the Minkowski embedding theorem and applications

Puri and Ralescu (1985) gave, recently, an extension of the Minkowski Embedding Theorem for the class double-struck E signⁿ_L of fuzzy sets u on ℝⁿ with the level application α → L_αu Lipschitzian on the C([0, 1] × S^n-1) space. In this work we extend the above result to the class double-struck E signⁿ_C of level-continuous applications. Moreover, we prove that double-struck E signⁿ_C is a complete metric space with double-struck E signⁿ_L⊈double-struck E signⁿ_C and double-struck E signⁿ_L = double-struck E signⁿ_C. 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).