Axiomatic and algebraic convexity of regular pairs
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 \(\mathcal{A}\)- notation is explained in [the author and \textit{B. Popovics}, ibid. 107, No. 1, 77--88 (2016; Zbl 1344.52002)]. From a different point of view, the point \(\left( x,y\right) \) in \(I\times\mathbb{R}\) is called an \(\omega \)-convex combination of \(n\) points \(\left( x_{1},y_{1}\right) ,\dots,\left( x_{n},y_{n}\right) \) in \(I\times\mathbb{R}\) with nonnegative coefficients \(\left( \lambda _{1},\dots,\lambda _{n}\right) \) if \[ \sum_{k=1}^{n}\lambda _{k}\omega \left( x_{k}\right) =\omega \left( x\right) \text{ and }\sum_{k=1}^{n}\lambda _{k}y_{k}=y. \] Notions as \(\omega \)-segments, \(\omega \)-convex sets and \(\omega \)-convex hulls are also considered. A set \(H\subset I\times\mathbb{R}\) is called \(\omega \)-convex if it is closed under \(\omega \)-convex combinations. The \(\omega \)-convex hull of \(H\) is the set of all \(\omega \) -convex combinations of the elements of \(H\). The main result of the paper states that the two approaches above are equivalent, that is the convex structures coincide.
scientific article; zbMATH DE number 6876769