Embedding of level-continuous fuzzy sets on Banach spaces

In this paper we present an extension of the Minkowski embedding theorem, showing the existence of an isometric embedding between the class ℱ_c(script x sign) of compact-convex and level-continuous fuzzy sets on a real separable Banach space (script x sign) and script c sign([0,1] × B(script x sign^*)), the Banach space of real continuous functions defined on the cartesian product between [0,1] and the unit ball B(script x sign^*) in the dual space script x sign^*. Also, by using this embedding, we give some applications to the characterization of relatively compact subsets of ℱ_c(script x sign). In particular, an Ascoli-Arzelá type theorem is proved and applied to solving the Cauchy problem ẋ(t) = f(t,x(t)), x(t₀) = x₀ on ℱ_c(script x sign).