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2021 Max-plus convexity in Archimedean Riesz spaces
Charles Horvath
Topol. Methods Nonlinear Anal. Advance Publication 1-43 (2021). DOI: 10.12775/TMNA.2020.028

Abstract

We study the topological properties of max-plus convex sets in an Archimedean Riesz space $E$ with respect to the topology and the max-plus structure associated to a given order unit $\boldsymbol u$; the definition of max-plus convex sets is algebraic and we do not assume that $E$ has an a priori given topological structure. To a given unit $\boldsymbol{u}$ one can associate two equivalent norms on $E$ one of which, denoted $\|\cdot\|_{\boldsymbol u}$, is classical, the other $\|\cdot\|_{\boldsymbol{hu}}$ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance ${\sf D}_{\boldsymbol h\boldsymbol u}$ on $E$ associated to $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ is a geodesic distance, called the Hilbert affine distance associated to $\boldsymbol u$, for which max-plus convex sets in $E$ are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and continuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex $\Delta_n$ equipped with the classical Hilbert metric-defined by the cross-ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is $\mathbb R^n$ with the Hilbert affine norm $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ with respect to $\boldsymbol u = (1, \ldots, 1)$.

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Charles Horvath. "Max-plus convexity in Archimedean Riesz spaces." Topol. Methods Nonlinear Anal. Advance Publication 1 - 43, 2021. https://doi.org/10.12775/TMNA.2020.028

Information

Published: 2021
First available in Project Euclid: 3 June 2021

Digital Object Identifier: 10.12775/TMNA.2020.028

Rights: Copyright © 2021 Juliusz P. Schauder Centre for Nonlinear Studies

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