Geodesic
Ky Fan's Lemma for Metric Spaces and an Approximation to the Goldbach's Problem
Journal of convex analysis, Tome 31 (2024) no. 1, pp. 265-277
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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.
Classification : 52A40, 11P32
Mots-clés : Metric space, d-convexity, Ky Fan's Lemma, Goldbach's problem 
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     author = {O. Galdames-Bravo},
     title = {Ky {Fan's} {Lemma} for {Metric} {Spaces} and an {Approximation} to the {Goldbach's} {Problem}},
     journal = {Journal of convex analysis},
     pages = {265--277},
     year = {2024},
     volume = {31},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2024_31_1_JCA_2024_31_1_a15/}
}
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O. Galdames-Bravo. Ky Fan's Lemma for Metric Spaces and an Approximation to the Goldbach's Problem. Journal of convex analysis, Tome 31 (2024) no. 1, pp. 265-277. http://geodesic.mathdoc.fr/item/JCA_2024_31_1_JCA_2024_31_1_a15/