About convex structures on metric spaces

"In the present paper we study the relationships between different concepts of convex structures in metric spaces that that are related to the works of K. Menger [Menger, K. Untersuchungen \""{u}ber allgemeine Metrik. {\it Math. Ann.} {\bf100} (1928), 75--163], H. Busemann [Busemann, H. {\it The geometry of geodesics}, Academic Press, 1955], I. N. Herstein; J. Milnor [Herstein, I. N.; Milnor, J. An axiomatic approach to measurable utility. {\it Econometrica} {\bf21} (1953), 291--297], E. Michael [Michael, E. Convex structures and continuous selections. {\it Canad. J. Math.} {\bf11} (1959), 556--575], A. Nijenhuis [Nijenhuis, A. A note on hyperconvexity in Riemannian manifolds. {\it Canad. J. Math.} {\bf11} (1959), 576--582.], W. Takahashi; T. Shimizu [Shimizu, T.; Takahashi, W. Fixed points of multivalued mappings in certain metric spaces. {\it Topol. Methods Nonlinear Anal.} {\bf8} (1996), no. 1, 197--203 and Takahashi, W. A convexity in metric space and nonexpansive mappings I. {\it Kodai Math. Sem. Rep.} {\bf 22} (1970), 142--149], M. Taskovi\'{c} [Taskovi\'{c}, M. General convex topological spaces and fixed points. {\it Math. Moravica} {\bf 1} (1997), 127--134], Yu. A. Aminov [Aminov, Yu. A. Two-Dimensional Surfaces in 3-Dimensional and 4-Dimensional Euclidean Spaces. Results and Unsolved Problems. {\it Ukr. Math. J.} {\bf 71} (2019), no. 1, 1--38.], H. V. Machado [Machado, H. V. A characterization of convex subsets of normed spaces.{\it Kodai Math. Sem. Rep.} {\bf25} (1973), 307--320], and many other papers. Some well known examples of concrete convex structures are reexamined and the possibilities of different embeddings of metric spaces with convex structures are also studied. Corollary \ref{C5.1} states that the Bolyai-Lobachevskii plane and the Bolyai-Lobachevskii half-plane are not isometrically embedding in some strictly convex normed space. A characteristic of the invariant metric generated by a norm is presented (Proposition \ref{P4.1}). "