Let X be a fuzzy set--valued random variable (\frv{}), and \huku{X} the
family of all fuzzy sets B for which the Hukuhara difference X\HukuDiff B
exists P--almost surely. In this paper, we prove that X can be
decomposed as X(\omega)=C\Mink Y(\omega) for P--almost every
ω∈Ω, C is the unique deterministic fuzzy set that minimizes
E[d2(X,B)2] as B is varying in \huku{X}, and Y is a centered
\frv{} (i.e. its generalized Steiner point is the origin). This decomposition
allows us to characterize all \frv{} translation (i.e. X(\omega) = M \Mink
\indicator{\xi(\omega)} for some deterministic fuzzy convex set M and some
random element in \Banach). In particular, X is an \frv{} translation if
and only if the Aumann expectation EX is equal to C up to a
translation.
Examples, such as the Gaussian case, are provided.Comment: 12 pages, 1 figure. v2: minor revision. v3: minor revision;
references, affiliation and acknowledgments added. Submitted versio