# Fuzzy vectors via convex bodies

## TL;DR

This paper introduces a convex geometric framework for fuzzy vectors, representing them via support functions, and explores their properties including symmetric, skew, and Mareš cores, with a focus on the uniqueness of cores in different dimensions.

## Contribution

It provides a novel convex-geometric representation of fuzzy vectors using support functions and characterizes the Mareš core, including its uniqueness in one-dimensional cases.

## Key findings

- Support functions characterize fuzzy vectors.
- Symmetric and skew fuzzy vectors are defined.
- Unique Mareš core exists only in one dimension.

## 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 $n=1$.