Convex space

A convex space can be defined as a set ${\displaystyle X}$ equipped with a binary convex combination operation ${\displaystyle c_{\lambda }:X\times X\rightarrow X}$ for each ${\displaystyle \lambda \in [0,1]}$ satisfying:

• ${\displaystyle c_{0}(x,y)=x}$
• ${\displaystyle c_{1}(x,y)=y}$
• ${\displaystyle c_{\lambda }(x,x)=x}$
• ${\displaystyle c_{\lambda }(x,y)=c_{1-\lambda }(y,x)}$
• ${\displaystyle c_{\lambda }(x,c_{\mu }(y,z))=c_{\lambda \mu }\left(c_{\frac {\lambda (1-\mu )}{1-\lambda \mu }}(x,y),z\right)}$ (for ${\displaystyle \lambda \mu \neq 1}$)

From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple ${\displaystyle (\lambda _{1},\dots ,\lambda _{n})}$, where ${\displaystyle \sum _{i}\lambda _{i}=1}$.

Any real affine space is a convex space. More generally, any convex subset of a real affine space is a convex space.

Convex spaces have been independently invented many times and given different names, dating back at least to Stone (1949).^[3] They were also studied by Neumann (1970)^[4] and Świrszcz (1974),^[5] among others.

Herstein and Milnor (1953)^[6] used convex spaces to prove the Mixture-space theorem.