Affine Hull

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,

:$\operatorname (S)=\left\.$

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 $S, T \subseteq X$

* $\operatorname(\operatorname S) = \operatorname S$
* $\operatorname S$ is a closed set if $X$ is finite dimensional.
* $\operatorname(S + T)=\operatorname S + \operatorname T$
* If $0 \in S$ then $\operatorname S = \operatorname S$.
* If $s_0 \in S$ then $\operatorname(S) - s_0 = \operatorname(S - s_0)$ is a linear subspace of $X$.
* $\operatorname(S - S) = \operatorname(S - S)$.
** So in particular, $\operatorname(S - S)$ is always a vector subspace of $X$.
* If $S$ is convex then $\operatorname(S - S) = \displaystyle\bigcup_ \lambda (S - S)$
* For every $s_0 \in S$, $\operatorname S = s_0 + \operatorname(S - S)$ where $\operatorname(S - S)$ is the smallest cone containing $S - S$ (here, a set $C \subseteq X$ is a cone if $r c \in C$ for all $c \in C$ and all non-negative $r \geq 0$).
** Hence $\operatorname(S - S)$ is always a linear subspace of $X$ parallel to $\operatorname S$.

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.

Affine geometry
Closure operators