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.^[1][2]

Formal Definition

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 ${\displaystyle \sum _{i}\lambda _{i}=1}$ .

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).^[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.

References

1. "Convex space". nLab. Retrieved 3 April 2023.
2. Fritz, Tobias (2009). "Convex Spaces I: Definition and Examples". arXiv:0903.5522 [math.MG].
3. Stone, Marshall Harvey (1949). "Postulates for the barycentric calculus". Annali di Matematica Pura ed Applicata. 29: 25–30. doi:10.1007/BF02413910. S2CID 122252152.
4. Neumann, Walter David (1970). "On the quasivariety of convex subsets of affine spaces". Archiv der Mathematik. 21: 11–16. doi:10.1007/BF01220869. S2CID 124051153.
5. Świrszcz, Tadeusz (1974). "Monadic functors and convexity". Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques. 22: 39–42.
6. Herstein, Israel Nathan; Milnor, John (1953). "An Axiomatic Approach to Measurable Utility". Econometrica. 21 (2): 291–297. doi:10.2307/1905540. JSTOR 1905540.