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

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.