
Journal of Convex Analysis 31 (2024), No. 1, 265277 Copyright Heldermann Verlag 2024 Ky Fan's Lemma for Metric Spaces and an Approximation to the Goldbach's Problem Orlando GaldamesBravo Departament de Matematiques, CIPFP Vicente Blasco Ibanez, Valencia, Spain galdames@uv.es [Abstractpdf] Given a metric space $(X,d)$ and a subset $K\subseteq X$ we say $K$ is $d$convex if for every $x,y\in K$, the segment between them defined as\\[1mm] \centerline{$[x,y]:=\{z\in X: d(x,y)=d(x,z)+d(z,y)\}$}\\[1mm] satisfy $[x,y]\subseteq K$. We generalize this notion to subsets where this condition is satisfied for a subset of segments that cover the subset. Then we show versions of a Ky Fan's Lemma on spaces with this property. As an application, we introduce an approximation to the Goldbach's problem. Keywords: Metric space, dconvexity, Ky Fan's Lemma, Goldbach's problem. MSC: 52A40; 11P32. [ Fulltextpdf (129 KB)] for subscribers only. 