Tropical convexity over max-min semiring

TL;DR

This paper surveys tropical convexity over the max-min semiring, introduces new results including colorful Caratheodory theorems, and discusses extensions of classical convexity theorems in a max-T semiring context.

Contribution

It provides a comprehensive overview of max-min tropical convexity and introduces novel colorful Caratheodory theorems and extensions of Radon and Tverberg theorems.

Findings

New colorful max-min Caratheodory theorems

Extensions of Radon and Tverberg theorems to max-T semirings

Descriptions of max-min segments, semispaces, and hyperplanes

Abstract

This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new "colorful" extensions of the max-min Caratheodory theorem. In the end of the paper, we list some consequences of the topological Radon and Tverberg theorems (like Helly and Centerpoint theorems), valid over a more general class of max-T semirings, where multiplication is a triangular norm.