Convex space
In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any finite set 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} .