Max-plus convexity in Archimedean Riesz spaces - Horvath, Charles (Topological Methods in Nonlinear Analysis, 2021)
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 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 one can associate two equivalent norms on E one of which, denoted \|·\|_u, is classical, the other \|·\|_ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance D_hu on E associated to \|·\|_hu is a geodesic distance, called the Hilbert affine distance associated to 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 Δ_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 R^n with the Hilbert affine norm \|·\|_hu with respect to u = (1, …, 1).