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 equipped with a binary convex combination operation for each satisfying:

  • (for )

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

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

  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.
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