Fuzzy vectors via convex bodies

Abstract

In the most accessible terms this paper presents a convex-geometric approach to the study of fuzzy vectors. Motivated by several key results from the theory of convex bodies, 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. As applications, 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, and 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 $n=1$.