Criteria for approximation of linear and affine functions
type: Item
instance of: scholarly article
title: Criteria for approximation of linear and affine functions
zbMATH Open document ID: 0566.52003
DOI: 10.1007/BF01200469
author: Christian Ronse
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: We consider a subset S of \({\mathfrak R}^ m\) and we suppose that to each \(x\in S\) is associated a convex subset C(x) of \({\mathfrak R}^ k\); we give then a criterion for the existence of a linear or affine function \(f: {\mathfrak R}^ m\to {\mathfrak R}^ k\) such that for every \(x\in S\), f(x)\(\in C(x)\). This contains as particular cases Helly's Theorems, the Transversal Theorem of Santaló, and some results of Grünbaum and Karlin-Shapley. We concentrate then on the case \(k=1\) and derive criteria for a function \(g: S\to {\mathfrak Z}\) to be the approximation of a linear or affine function \(f: S\to {\mathfrak R}\). We show how this is related to the definition of straightness in digital geometry.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 3903452
zbMATH Keywords: linear functions
zbMATH Keywords: affine functions
zbMATH Keywords: convex sets
zbMATH Keywords: digitalization
zbMATH Keywords: Helly's theorems
zbMATH Keywords: transversal theorem of Santaló
zbMATH Keywords: Grünbaum
zbMATH Keywords: Karlin-Shapley
MaRDI profile type: Publication
cites work: Common Transversals for Families of Sets†
cites work: Some applications of a theorem on convex functions
cites work: Three-Dimensional Digital Planes
cites work: A simple proof of Rosenfeld's characterization of digital straight line segments
cites work: Digital Straight Line Segments
cites work: Q5603731
description: scientific article; zbMATH DE number 3903452