On starshaped fuzzy sets (Q1037964)

A fuzzy set \(\mu\) in \(\mathbb R^n\) is said to be starshaped relative to \(y \in\mathbb R^n\) if for all \(x \in\mathbb R^n\) there holds \(\mu(\lambda (x-y) + y) \geq \mu (x)\), \(0\leq\lambda \leq 1\). Relationships of being starshaped with other concepts of starshapedness are studied, namely: starshapedness of all the level sets, fulfilling the inequality \(\mu(\lambda x + (1-\lambda )y) \geq \min \{\mu (x), \mu (y) \}\) or \(\mu(\lambda x + (1-\lambda )y) \geq \lambda \mu (x), + (1-\lambda )\mu (y)\). The properties of starshaped sets and their shadows are discussed.