Hyperspaces of MAX-plus convex subsets of powers of the real line
Let \(\tau\) be a cardinal number, \(x,y\in\mathbb R^{\tau}\), \(\lambda\in\mathbb R\), and denote \(x\oplus y\) the coordinatewise maximum of \(x\) and \(y\), and \(\lambda \odot x\) the vector obtained from \(x\) by adding \(\lambda\) to each of its coordinates. A set \(A\subseteq \mathbb R^{\tau}\) is \textit{max-plus convex}, provided \(\alpha\odot a\oplus \beta\odot b\in A\) for all \(a,b\in A\) and \(\alpha,\beta \in\mathbb R\cup\{-\infty\}\) with \(\alpha\oplus \beta=0\). The main result of the paper is that the hyperspace \(\text{mpcc}(\mathbb R^{\tau})\) of the nonempty max-plus convex compact subsets of \(\mathbb R^{\tau}\) endowed with the Hausdorff metric topology is homeomorphic to \(\mathbb R^{\tau}\) if \(\tau\in \{\omega,\omega_1\}\); moreover, it is shown that \(\text{mpcc}(\mathbb R^{\tau})\) is not an absolute retract if \(\tau > \omega_1\). The paper ends with two open problems.
scientific article; zbMATH DE number 6062837
scientific article