Maxplus convexity in Riesz spaces
Abstract
We study maxplus convexity in an Archimedean Riesz space $E$ with an order unit $\un$; the definition of maxplus convex sets is algebraic and we do not assume that $E$ has an {\it a priori} given topological structure. To the given unit $\un$ one can associate two equivalent norms $\norm\cdot\norm_{\un}$ and $\norm\cdot\norm_{\hun}$ on $E$; the distance ${\sf D}_{\hun}$ on $E$ associated to $\norm\cdot\norm_{\hun}$ is a geodesic distance for which maxplus convex sets in $E$ are geodesically closed sets. Under suitable assumptions, we establish maxplus versions of some fixed points and continuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact maxplus convex sets are Absolute Retracts.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.00946
 Bibcode:
 2019arXiv190500946H
 Keywords:

 Mathematics  Functional Analysis;
 14T99;
 46A40;
 54H25;
 54C55;
 54C65;
 62P20
 EPrint:
 20 pages, 9 figures