Busemann G-space

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

If is a metric space such that

1. for every two distinct there exists such that (Menger convexity)
2. every -bounded set of infinite cardinality possesses accumulation points
3. for every there exists such that for any distinct points there exists such that (geodesics are locally extendable)
4. for any distinct points , if such that , and , then (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.

References