Betweenness for real vectors and lines. III: Alternative characterizations of betweennesses (Q1057299)

This paper continues the authors' work on betweenness [ibid. 30, 125-162 and 443-464 (1978; Zbl 0462.06005 and Zbl 0462.06006)] and completes the discussion of this concept for real vector spaces. Initially, a new terminology enabling the discussion of center-related betweennesses defined with respect to centers other than the origin is introduced and it is proved that the \({\mathcal V}^ v_{\nu}\)-betweenness in \({\mathcal O}_{{\mathcal K}}\), which is origin-centered, is equivalent to a non- origin-centered \({\mathcal V}^ v_ 0\)-betweenness in one lower dimension. Moreover, a new betweenness in \({\mathcal K}\) is defined and studied, and it is shown that its restriction to \({\mathcal O}_{{\mathcal K}}\) is precisely \({\mathcal V}^ v_{\nu}\)-betweenness for a certain \(\nu\). And finally, another new betweenness is induced on the upper open-hemisphere-plus-a- point in 3-space, and a characterization of it is obtained which is expected to facilitate later investigations of betweenness in complex spaces.