Busemann G-space

In mathematics, a Busemann G-space is a type of metric space first described by Herbert Busemann in 1942.

If [math]\displaystyle{ (X,d) }[/math] is a metric space such that

1. for every two distinct [math]\displaystyle{ x, y \in X }[/math] there exists [math]\displaystyle{ z \in X-\{x,y\} }[/math] such that [math]\displaystyle{ d(x,z)+d(y,z)=d(x,z) }[/math] (Menger convexity)
2. every [math]\displaystyle{ d }[/math]-bounded set of infinite cardinality possesses accumulation points
3. for every [math]\displaystyle{ w \in X }[/math] there exists [math]\displaystyle{ \rho_w }[/math] such that for any distinct points [math]\displaystyle{ x,y \in B(w,\rho_w) }[/math] there exists [math]\displaystyle{ z \in ( b(w,\rho_w)-\{ x,y \} )^\circ }[/math] such that [math]\displaystyle{ d(x,z)+d(y,z)=d(x,z) }[/math] (geodesics are locally extendable)
4. for any distinct points [math]\displaystyle{ x,y \in X }[/math], if [math]\displaystyle{ u,v \in X }[/math] such that [math]\displaystyle{ d(x,u)+d(y,u)=d(x,u) }[/math], [math]\displaystyle{ d(x,v)+d(y,v)=d(x,v) }[/math] and [math]\displaystyle{ d(y,u)=d(y,v) }[/math] (geodesic extensions are unique).

then X is said to be a Busemann G-space. Every Busemann G-space is a homogenous space.

The Busemann conjecture states that every Busemann G-space is a topological manifold. It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4.