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Seven-dimensional cross product

Affine, conical, and convex combinations[edit]

By restricting the coefficients used in linear combinations, one can define the related concepts of affine combinationconical combination, and convex combination, and the associated notions of sets closed under these operations.

Type of combinationRestrictions on coefficientsName of setModel space
Linear combinationno restrictionsVector subspace{\displaystyle \mathbf {R} ^{n}}
Affine combination{\displaystyle \sum a_{i}=1}Affine subspaceAffine hyperplane
Conical combination{\displaystyle a_{i}\geq 0}Convex coneQuadrant or Octant
Convex combination{\displaystyle a_{i}\geq 0} and {\displaystyle \sum a_{i}=1}Convex setSimplex

Because these are more restricted operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone.

These concepts often arise when one can take certain linear combinations of objects, but not any: for example, probability distributions are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and positive measures are closed under conical combination but not affine or linear – hence one defines signed measures as the linear closure.

Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of “positive”, and hence can only be defined over an ordered field (or ordered ring), generally the real numbers.

If one allows only scalar multiplication, not addition, one obtains a (not necessarily convex) cone; one often restricts the definition to only allowing multiplication by positive scalars.

All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as “vector spaces forgetting the origin”), rather than being axiomatized independently.