- Source: Convex space
In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points.
Formal Definition
A convex space can be defined as a set
X
{\displaystyle X}
equipped with a binary convex combination operation
c
λ
:
X
×
X
→
X
{\displaystyle c_{\lambda }:X\times X\rightarrow X}
for each
λ
∈
[
0
,
1
]
{\displaystyle \lambda \in [0,1]}
satisfying:
c
0
(
x
,
y
)
=
x
{\displaystyle c_{0}(x,y)=x}
c
1
(
x
,
y
)
=
y
{\displaystyle c_{1}(x,y)=y}
c
λ
(
x
,
x
)
=
x
{\displaystyle c_{\lambda }(x,x)=x}
c
λ
(
x
,
y
)
=
c
1
−
λ
(
y
,
x
)
{\displaystyle c_{\lambda }(x,y)=c_{1-\lambda }(y,x)}
c
λ
(
x
,
c
μ
(
y
,
z
)
)
=
c
λ
μ
(
c
λ
(
1
−
μ
)
1
−
λ
μ
(
x
,
y
)
,
z
)
{\displaystyle c_{\lambda }(x,c_{\mu }(y,z))=c_{\lambda \mu }\left(c_{\frac {\lambda (1-\mu )}{1-\lambda \mu }}(x,y),z\right)}
(for
λ
μ
≠
1
{\displaystyle \lambda \mu \neq 1}
)
From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple
(
λ
1
,
…
,
λ
n
)
{\displaystyle (\lambda _{1},\dots ,\lambda _{n})}
, where
∑
i
λ
i
=
1
{\displaystyle \sum _{i}\lambda _{i}=1}
.
Examples
Any real affine space is a convex space. More generally, any convex subset of a real affine space is a convex space.
History
Convex spaces have been independently invented many times and given different names, dating back at least to Stone (1949). They were also studied by Neumann (1970) and Świrszcz (1974), among others.
References
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