A characterization of $2$-dimensional spherical space

Abstract:

The midset of two distinct points $a$ and $b$ of a metric space is defined as the set of all points $x$ in the space for which the distances $ax$ and $bx$ are equal. A metric space is said to have the $1$-WLMP if the midset of each two distinct points is a convex $1$-sphere having the property that each nonmaximal (with respect to inclusion) segment intersecting it twice lies in it. We show that a nondegenerate compact space $X$ is isometric to a $2$-dimensional spherical space ${S_{2,\alpha }}$ (a $2$-dimensional sphere of radius $\alpha$ in euclidean $3$-space with the “shorter arc” metric) if and only if $X$ has a metric with the $1$-WLMP.