Convex space explained

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.^{[1] [2]}

Formal Definition

A convex space can be defined as a set

X equipped with a binary convex combination operation c _{λ} : X x X → X for each λ \in [ 0 , 1 ] satisfying:c _{0(x,y)=x} c _{1(x,y)=y} c _{λ(x,x)=x} c _{λ(x,y)=c} _{1-λ} (y,x) c _{λ(x,c} _{\mu(y,z))=c} _{λ\mu} \left(c _{(λ ( 1- \mu))/(1-λ \mu)} (x,y),z\right) (for λ \mu ≠ 1 )

From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple

( λ _{1,} ... , λ _{n)} , where \sum _{iλ} _{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.

Notes and References

1. Web site: Convex space . nLab . 3 April 2023.
2. Fritz . Tobias . Convex Spaces I: Definition and Examples . 2009 . math.MG . 0903.5522 .
3. Stone . Marshall Harvey . Postulates for the barycentric calculus . Annali di Matematica Pura ed Applicata . 1949 . 29 . 25–30. 10.1007/BF02413910 . 122252152 .
4. Neumann . Walter David . On the quasivariety of convex subsets of affine spaces . Archiv der Mathematik . 1970 . 21 . 11–16. 10.1007/BF01220869 . 124051153 .
5. Świrszcz . Tadeusz . Monadic functors and convexity . Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques . 1974 . 22 . 39–42.