The structure of max-plus hyperplanes
type: Item
instance of: scholarly article
title: The structure of max-plus hyperplanes
zbMATH Open document ID: 1130.52001
author: Ivan Singer
author: Viorel Niţică
published in: Linear Algebra and its Applications
publication date: 2007-09-14T00:00:00Z
review text: The authors study max-plus hyperplanes in the semimodule \(R^n_{\max}\) over the max-plus semifield \(R_{\max}\), where \(R_{\max}:= R\cup\{-\infty\}\) is with the operations \(\oplus:= \max\) and \(\otimes:= +\). From the authors' summary: We give a formula that expresses a nondegenerate strictly affine hyperplane as a union of complements of semispaces at a point \(z \in R^n\), called the ``center'' of \(H\), with the boundary of a union of complements of other semispaces at \(z\). Using this formula, we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior. We give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center, and we characterize the cases in which the boundary \(bd (H)\) of a nondegenerate strictly affine hyperplane \(H\) is also a hyperplane. Next, we give the relations between nondegenerate strictly affine hyperplanes \(H\), their centers \(z\), and their coefficients. In the converse direction we show that any union of complements of semispaces at a point \(z \in R^n\) with the boundary of any union of complements of some other semispaces at that point \(z\), is a nondegenerate strictly affine hyperplane. We obtain a formula for the total number of strictly affine hyperplanes. We give complete lists of all strictly affine hypeprlanes for the cases \(n=1\) and \(n=2\). We show that each linear hyperplane \(H\) in \(R_{\max}^n\) can be decomposed as the union of four parts, where each part is easy to describe in terms of complements of semispaces, some of them in a lower dimensional space.
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 5190949
zbMATH Keywords: max-plus hyperplane
zbMATH Keywords: max-plus convex set
zbMATH Keywords: max-plus segment
zbMATH Keywords: max-plus semispace
reviewed by: Kelly J. Pearson
MaRDI profile type: Publication
full work available at URL: j.laa.2007.05.016
OpenAlex ID: W2091381132
cites work: Q4140769
cites work: Max-plus convex sets and max-plus semispaces. I
cites work: Max-plus convex sets and max-plus semispaces. II
cites work: Duality and separation theorems in idempotent semimodules.
cites work: Q3820367
cites work: Q5702473
cites work: Q5702484
DOI: 10.1016/J.LAA.2007.05.016
description: scientific article; zbMATH DE number 5190949