Affine maps that induce polyhedral complex isomorphisms
type: Item
instance of: scholarly article
title: Affine maps that induce polyhedral complex isomorphisms
zbMATH Open document ID: 0982.52013
author: Andreas W. M. Dress
author: Katharina T. Huber
author: Vincent Moulton
published in: Discrete \& Computational Geometry
publication date: 2000-08-24T00:00:00Z
review text: An affine map between two polyhedral complexes which is bijective on the underlying sets need not induce an isomorphism of the complexes preserving the induced cell-complex structure. The key observation in this paper is that this is nevertheless true under an additional assumption: If an affine bijection \(f: T_1\to T_2\) between two polyhedral complexes \(T_1, T_2\), both of which consist of a union of faces of two convex polyhedra \(P_1\), \(P_2\), extends to an affine map from \(P_1\) to \(P_2\) then \(f\) necessarily respects the cell-complex structure of \(T_1\) and \(T_2\) induced by \(P_1\) and \(P_2\), respectively. This result depends on certain properties of extremal points and extremal subsets of polyhedra. There are applications in the sciences, including biology and molecular evolution.
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 1495780
zbMATH Keywords: polyhedral complex isomorphisms
zbMATH Keywords: polyhedral cell
zbMATH Keywords: extremal points
zbMATH Keywords: \(T\)-theory
zbMATH Keywords: affine map
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: s004540010034
OpenAlex ID: W2091382594
DOI: 10.1007/S004540010034
description: scientific article; zbMATH DE number 1495780