On an odule in Hilbert geometry (Q1909840)

The notion of odular structure was introduced by \textit{L. V. Sabinin} as an algebraic abstraction derived from the existence of local normal coordinates in affinely connected manifolds [Sov. Math., Dokl. 18, 515-518 (1977); translation from Dokl. Akad. Nauk SSSR 233, 800-803 (1977; Zbl 0375.53021)]. The present article concerns a metric space \((X,\rho)\) in which two distinct points can be joined by a unique shortest line and in which each shortest line can be uniquely extended to a geodesic line. An odular structure \(W\) on \((X,\rho)\) is then defined by the author; it consists of mappings \(L^p : X \times X \to X\) and \(\omega^p : \mathbb{R} \times X \to X\) associated to the points \(p\) of \(X\); the structure \(W\) is called topological if \(L^p\) and \(\omega^p\) are continuous for every \(p \in X\). If \((X,\rho)\) is finitely compact then \(W\) is topological (Theorem 1). The examples developed by the author are: the Hilbert geometry of a ball in a strictly convex Banach space, and the Beltrami-Klein model of the Lobachevskij geometry in a ball belonging to a Hilbert space.