# Affine hull

In mathematics, the affine hull of a set ${\displaystyle S}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is the smallest affine set containing ${\displaystyle S}$, or equivalently, the intersection of all affine sets containing ${\displaystyle S}$. Here, an affine set may be defined as the translation of a vector subspace.

The affine hull ${\displaystyle {\text{aff}}(S)}$ of ${\displaystyle S}$ is the set of all affine combinations of elements of ${\displaystyle S}$, that is,

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

## Examples

• The affine hull of the empty set is the empty set.
• The affine hull of a singleton (a set made of one single element) is the singleton itself.
• 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 ${\displaystyle \mathbb {R} ^{3}}$ is the entire space ${\displaystyle \mathbb {R} ^{3}}$.

## Properties

• ${\displaystyle \mathrm {aff} (\mathrm {aff} (S))=\mathrm {aff} (S)}$
• ${\displaystyle \mathrm {aff} (S)}$ is a closed set
• ${\displaystyle \mathrm {aff} (S+F)=\mathrm {aff} (S)+\mathrm {aff} (F)}$
• If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all ${\displaystyle \alpha _{i}}$ 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 ${\displaystyle \alpha _{i}}$, 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.