Max-plus convexity in Archimedean Riesz spaces

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 {\it a priori} given topological structure. To a given unit $\un$ one can associate two equivalent norms on $E$ one of which, denoted $\|\cdot\|_{\boldsymbol u}$, is classical, the other $\|\cdot\|_{\hun}$ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance ${\sf D}_{\boldsymbol h\boldsymbol u}$ on $E$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 $\boldsymbol u = (1, \ldots, 1)$.