Hyperspaces of MAX-plus convex subsets of powers of the real line
type: Item
instance of: scholarly article
title: Hyperspaces of MAX-plus convex subsets of powers of the real line
zbMATH Open document ID: 1250.54011
author: Mykhailo Zarichnyi
author: Lidiya Bazylevych
author: Dušan Repovš
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.1949
review text: 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.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C55
zbMATH DE Number: 6062837
zbMATH Keywords: max-plus convex set
zbMATH Keywords: hyperspace
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: absolute retract
zbMATH Keywords: powers of the real line
reviewed by: Laszlo Zsilinszky
MaRDI profile type: Publication
OpenAlex ID: W2139224404
arXiv ID: 1206.1949
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DOI: 10.1016/J.JMAA.2012.05.002
description: scientific article; zbMATH DE number 6062837