In mathematics, the **affine hull** or **affine span** of a set *S* in Euclidean space **R**^{n} 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,

## 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
**R**^{3}is the entire space**R**^{3}.

## Properties

For any subsets

- is a closed set if is finite dimensional.
- If then .
- If then is a linear subspace of .
- .
- So in particular, is always a vector subspace of .

- If is convex then
- For every , where is the smallest cone containing (here, a set is a
**cone**if for all and all non-negative ).- Hence is always a linear subspace of parallel to .

## Related sets

- If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all 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 , 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.