Busemann G-space

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

If ${\displaystyle (X,d)}$ is a metric space such that

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

then X is said to be a Busemann G-space. Every Busemann G-space is a homogeneous 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.^[1][2]