Hyperspaces of max-plus convex subsets of powers of the real line

TL;DR

This paper characterizes the topological structure of hyperspaces of max-plus convex subsets in powers of the real line, showing they are absolute retracts only for certain cardinalities.

Contribution

It extends the concept of max-plus convexity to Tychonov powers and describes the topology of their hyperspaces, identifying conditions for them to be absolute retracts.

Findings

Hyperspaces are ARs if and only if the power's index is at most .

Provides a topological classification of max-plus convex hyperspaces.

Extends max-plus convexity concepts to infinite-dimensional settings.

Abstract

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^\tau$ of the real line. We show that the corresponding spaces are AR's if and only if $\tau\le\omega_1$.