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

for every two distinct $x, y \in X$ there exists $z \in X ∖ {x, y}$ such that $d (x, z) + d (y, z) = d (x, y)$ (Menger convexity)
every $d$ -bounded set of infinite cardinality possesses accumulation points
for every $w \in X$ there exists $ρ_{w}$ such that for any distinct points $x, y \in B (w, ρ_{w})$ there exists $z \in (B (w, ρ_{w}) ∖ {x, y})^{\circ}$ such that $d (x, y) + d (y, z) = d (x, z)$ (geodesics are locally extendable)
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 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]

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