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.[1][2]
Formal Definition
A convex space can be defined as a set [math]\displaystyle{ X }[/math] equipped with a binary convex combination operation [math]\displaystyle{ c_\lambda : X \times X \rightarrow X }[/math] for each [math]\displaystyle{ \lambda \in [0,1] }[/math] satisfying:
- [math]\displaystyle{ c_0(x,y)=x }[/math]
- [math]\displaystyle{ c_1(x,y)=y }[/math]
- [math]\displaystyle{ c_\lambda(x,x)=x }[/math]
- [math]\displaystyle{ c_\lambda(x,y)=c_{1-\lambda}(y,x) }[/math]
- [math]\displaystyle{ c_\lambda(x,c_\mu(y,z))=c_{\lambda\mu}\left(c_{\frac{\lambda(1-\mu)}{1-\lambda\mu}}(x,y),z\right) }[/math] (for [math]\displaystyle{ \lambda\mu\neq 1 }[/math])
From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple [math]\displaystyle{ (\lambda_1, \dots, \lambda_n) }[/math], where [math]\displaystyle{ \sum_i\lambda_i = 1 }[/math].
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.
References
- ↑ "Convex space". https://ncatlab.org/nlab/show/convex+space.
- ↑ Fritz, Tobias (2009). "Convex Spaces I: Definition and Examples". arXiv:0903.5522 [math.MG].
- ↑ Stone, Marshall Harvey (1949). "Postulates for the barycentric calculus". Annali di Matematica Pura ed Applicata 29: 25–30. doi:10.1007/BF02413910.
- ↑ Neumann, Walter David (1970). "On the quasivariety of convex subsets of affine spaces". Archiv der Mathematik 21: 11–16. doi:10.1007/BF01220869.
- ↑ Świrszcz, Tadeusz (1974). "Monadic functors and convexity". Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques 22: 39–42.
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