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Journal of Convex Analysis 31 (2024), No. 1, 265--277
Copyright Heldermann Verlag 2024

Ky Fan's Lemma for Metric Spaces and an Approximation to the Goldbach's Problem

Orlando Galdames-Bravo
Departament de Matematiques, CIPFP Vicente Blasco Ibanez, Valencia, Spain


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, d-convexity, Ky Fan's Lemma, Goldbach's problem.

MSC: 52A40; 11P32.

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