Affine hull

In mathematics, the affine hull 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,

:$operatorname\{aff\}\; (S)=left\{sum\_\{i=1\}^k\; alpha\_i\; x\_i\; Bigg\; |\; x\_iin\; S,\; ,\; alpha\_iin\; mathbb\{R\},\; ,\; sum\_\{i=1\}^k\; alpha\_i=1,\; k=1,\; 2,\; dots\; ight\}.$

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 R^"3" is the entire space R^"3".

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 must not 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 is bigger than the affine hull of "S".

References

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