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Busemann G-space

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

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

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

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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