- Source: 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
)
{\displaystyle (X,d)}
is a metric space such that
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)
every
d
{\displaystyle d}
-bounded set of infinite cardinality possesses accumulation points
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)
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
Kata Kunci Pencarian:
- Meteorit Murchison
- Isotop helium
- Busemann G-space
- Hyperbolic metric space
- Herbert Busemann
- Busemann function
- Hilbert's fourth problem
- Busemann–Petty problem
- Glossary of Riemannian and metric geometry
- Operation Paperclip
- Bing–Borsuk conjecture
- Synthetic geometry
