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.[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

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