Tropical convexity over max-min semiring

摘要

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 nonsepar-ation results based on semispaces. There are some new results. In particular, we give new "colorful" extensions of the max-min Carathéodory 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.