Current Open Questions in Complete Mixability

TL;DR

This paper reviews open questions in the theory of complete and joint mixability, highlighting recent results and making the topic accessible to a broad audience interested in distribution theory, optimization, and risk management.

Contribution

It presents a comprehensive list of open questions in complete and joint mixability and includes some new results, broadening understanding and research directions.

Findings

Identification of key open questions in mixability theory

Introduction of new results related to mixability

Discussion of applications in risk management and optimization

Abstract

Complete and joint mixability has raised considerable interest in recent few years, in both the theory of distributions with given margins, and applications in discrete optimization and quantitative risk management. We list various open questions in the theory of complete and joint mixability, which are mathematically concrete, and yet accessible to a broad range of researchers without specific background knowledge. In addition to the discussions on open questions, some results contained in this paper are new.