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

Given a metric space

and a subset

K\subseteq X K ⊆ X

we say

K K

is

d d

-convex if for every

x,y\in K x , y ∈ K

, the segment between them defined as

[x,y]:=\{z\in X: d(x,y)=d(x,z)+d(z,y)\} [ x , y ] : = { z ∈ X : d ( x , y ) = d ( x , z ) + d ( z , y ) }

satisfy

[x,y]\subseteq K [ x , y ] ⊆ 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.