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
for every two distinct there exists such that (Menger convexity) every -bounded set of infinite cardinality possesses accumulation points for every there exists such that for any distinct points there exists such that (geodesics are locally extendable) 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.
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