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Convex space

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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.

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 } 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} .

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). They were also studied by Neumann (1970) and Świrszcz (1974), among others.

References

  1. "Convex space". nLab. Retrieved 3 April 2023.
  2. Fritz, Tobias (2009). "Convex Spaces I: Definition and Examples". arXiv:0903.5522 .
  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.
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