Affine hull

In mathematics, the affine hull of a set S in Euclidean space Rⁿ is the smallest affine set containing S, or equivalently, the 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,

<math>\operatorname{aff} (S)=\left\{\sum_{i=1}^k \alpha_i x_i \Bigg | x_i\in S, \, \alpha_i\in \mathbb{R}, \, \sum_{i=1}^k \alpha_i=1, k=1, 2, \dots\right\}.</math>

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

• <math>\mathrm{aff}(\mathrm{aff}(S)) = \mathrm{aff}(S)</math>
• <math>\mathrm{aff}(S)</math> 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 <math>\alpha_i</math> be non-negative, one obtains the convex hull of S, which must be smaller than the affine hull of S as more restrictions are involved. If however one puts no restrictions at all on the numbers <math>\alpha_i</math>, instead of an affine combination one has a linear combination, and the resulting set is the linear span of S, which is bigger than the affine hull of S.

References

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