TL;DR
This survey explores the theory of choice functions, focusing on their image and domain properties, and introduces new results on scrambled, weighted, and cooperative variants of rainbow sets.
Contribution
It provides a comprehensive overview of rainbow sets and introduces new theorems on scrambled, weighted, and cooperative choice functions.
Findings
Theorems on scrambled rainbow sets.
Results on weighted choice functions.
Findings on cooperative choice functions.
Abstract
This is a survey paper on rainbow sets (another name for ``choice functions''). The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed up set of sets) image, and those that have a large domain and small image, where ``smallness'' means belonging to some specified complex (a closed-down set). The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions - to be defined below.
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Taxonomy
TopicsFunctional Equations Stability Results · Optimization and Variational Analysis