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 (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.[1][2]

References

  1. M., Halverson, Denise; Dušan, Repovš (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). arXiv:0811.0886. ISSN 1331-0623.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  2. Papadopoulos, Athanase (2005). Metric Spaces, Convexity and Nonpositive Curvature. European Mathematical Society. p. 77. ISBN 9783037190104.


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