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 $z\in X-\{x,y\}$ such that $d(x,z)+d(y,z)=d(x,z)$ (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 $z\in (b(w,{\rho }_w)-\{x,y\}{)}^{\circ }$ such that $d(x,z)+d(y,z)=d(x,z)$ (geodesics are locally extendable)
4. for any distinct points $x,y\in X$, if $u,v\in X$ such that $d(x,u)+d(y,u)=d(x,u)$, $d(x,v)+d(y,v)=d(x,v)$ and $d(y,u)=d(y,v)$ (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.