Affine hull

In mathematics, the affine hull of a set S in Euclidean space Rⁿ is the smallest affine set containing S, or equivalrsection (set theory)|intersection]] of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace.

The affine hull aff(S) of S is the set of all affine combinations of elements of S, that is,

$\operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}{\Bigg |}k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.$

Examples

• The affine hull of a set of two different points is the line through them.
• The affine hull of a set of three points not on one line is the plane going through them.
• The affine hull of a set of four points not in a plane in R³ is the entire space R³.

Properties

• $\mathrm {aff} (\mathrm {aff} (S))=\mathrm {aff} (S)$
• $\mathrm {aff} (S)$ is a closed set

Related sets

• If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all $\alpha _{i}$ be non-negative, one obtains the convex hull of S, which cannot be larger than the affine hull of S as more restrictions are involved.
• The notion of conical combination gives rise to the notion of the conical hull
• If however one puts no restrictions at all on the numbers $\alpha _{i}$, instead of an affine combination one has a linear combination, and the resulting set is the linear span of S, which contains the affine hull of S.

References

• R.J. Webster, Convexity, Oxford University Press, 1994. ISBN 0-19-853147-8.