Affine hull

In mathematics, the affine hull of a set $S$ in Euclidean space $\mathbb {R} ^{n}$ 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 ${\text{aff}}(S)$ 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 |}\,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 $\mathbb {R} ^{3}$ is the entire space $\mathbb {R} ^{3}$ .

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.