It is known that every fuzzy number has a unique Mare\v{s} core and can be
decomposed in a unique way as the sum of a skew fuzzy number, given by its
Mare\v{s} core, and a symmetric fuzzy number. The aim of this paper is to
provide a negative answer to the existence of an $n$-dimensional version of the
above theorem. By applying several key tools from convex geometry, we establish
a representation theorem of fuzzy vectors through support functions, in which a
necessary and sufficient condition for a function to be the support function of
a fuzzy vector is provided. Futhermore, symmetric and skew fuzzy vectors are
postulated, based on which a Mare\v{s} core of each fuzzy vector is constructed
through convex bodies and support functions. It is shown that every fuzzy
vector over the $n$-dimensional Euclidean space has a unique Mare\v{s} core if,
and only if, the dimension $n=1$.