On the Mare\v{s} cores of fuzzy vectors

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 $n=1$.