Axiomatic and algebraic convexity of regular pairs
type: Item
instance of: scholarly article
title: Axiomatic and algebraic convexity of regular pairs
zbMATH Open document ID: 1402.52001
author: Mihály Bessenyei
published in: Journal of Geometry
publication date: 2018-05-30T00:00:00Z
review text: A continuous map \(\omega :I\rightarrow\mathbb{R}^{2},\) where \(I\) is an open interval, is said to be a regular pair, if for all elements \(x_{1}\), \(x_{2}\) of \(I\), the determinant\(\;\det (\omega (x_{1})\) \(\omega (x_{2}))\) is positive. As it is usually the case, such so called two-dimensional Chebyshev systems are interpolation families: Let \(p_{1}=\left( x_{1},y_{1}\right) \) and \( p_{2}=\left( x_{2},y_{2}\right) \) in \(I\times\mathbb{R}\), with pairwise distinct first coordinates. Then there exists a unique linear combination \(\ell _{12}\) of the components of \(\omega \) whose graph contains \(p_{1}\) and \(p_{2}\). This way it is possible to introduce a geometry on \(I\times\mathbb{R}\): the generalized lines are defined as the elements of the linear hull \(\mathrm{Lin}\left( \omega \right) \) of the components of \(\omega \) and the Euclidean vertical lines. The Euclidean geometry corresponds to the particular case \(\omega =\left( 1,\mathrm{id}\right) \). In a similar manner to the Euclidean definitions, notions as generalized segments, convex sets and convex hulls are defined in this framework: A generalized segment with endpoints \(p_{1}=\left( x_{1},y_{1}\right) \) and \( p_{2}=\left( x_{2},y_{2}\right) \) is \[ \left[ p_{1},p_{2}\right] :=\left\{ \left( x,y\right) \in I\times\mathbb{R}:x\in \left[ \min \left\{ x_{1},x_{2}\right\} ,\max \left\{ x_{1},x_{2}\right\} \right] ,y=\ell _{12}\left( x\right) \right\} , \] if \(x_{1}\neq x_{2}.\) Otherwise \[ \left[ p_{1},p_{2}\right] :=\left\{ \left( x_{1},y\right) \in I\times\mathbb{R}:y\in \left[ \min \left\{ y_{1},y_{2}\right\} ,\max \left\{ y_{1},y_{2}\right\} \right] \right\} . \] A set \(H\subset I\times\mathbb{R}\) is called \(\mathcal{A}\)-convex if \(\left[ p_{1},p_{2}\right] \subset H\) for all \(p_{1},p_{2}\in H.\) The \(\mathcal{A}\)-convex hull of \(H\) is the smallest \(\mathcal{A}\)-convex set which contains \(H,\) \(\mathrm{conv}_{ \mathcal{A}}\left( H\right) \). The \(\mathcal{A}\)- n…
reviewed by: F-C. Mitroi
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 6876769
zbMATH Keywords: convex structure
zbMATH Keywords: planar convexity
zbMATH Keywords: Chebyshev system
MaRDI profile type: Publication
full work available at URL: s00022-018-0429-z
OpenAlex ID: W2795311711
cites work: Q4301761
cites work: Q4790110
cites work: Generalized convex functions
cites work: Q2974985
cites work: Hadamard-type inequalities for generalized convex functions
cites work: Convexity without convex combinations
cites work: Convex structures induced by Chebyshev systems
cites work: Convex separation by regular pairs
cites work: Q5534404
cites work: Generalized convex sets.
cites work: Approximately generalized convex functions
cites work: Convex functions and their applications. A contemporary approach
cites work: Q5296430
cites work: A sandwich theorem and Hyers-Ulam stability of affine functions
cites work: Q3141898
DOI: 10.1007/S00022-018-0429-Z
description: scientific article; zbMATH DE number 6876769