Busemann G-space

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

If $$(X,d)$$ is a metric space such that


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