Affine hull

Short description: Smallest affine subspace that contains a subset

In mathematics, the affine hull or affine span 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 of $S$ is what $\mathrm{span}S$ would be if the origin was moved to $S$.

The affine hull aff($S$) of $S$ is the set of all affine combinations of elements of $S$, that is,

$\mathrm{aff}(S)=\left\{\sum _{i=1}^k{\alpha }_i{x}_i\right|k>0,x_i\in S,{\alpha }_i\in \mathbb{ℝ},\sum _{i=1}^k{\alpha }_i=1\}.$

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 ${\mathbb{ℝ}}^3$.

Properties

For any subsets $S,T\subseteq X$

• $\mathrm{aff}(\mathrm{aff}S)=\mathrm{aff}S\subset \mathrm{span}S=\mathrm{span}\mathrm{aff}S$.
• $\mathrm{aff}S$ is a closed set if $X$ is finite dimensional.
• $\mathrm{aff}(S+T)=\mathrm{aff}S+\mathrm{aff}T$.
• $S\subset \mathrm{aff}S$.
• If $0\in \mathrm{aff}S$ then $\mathrm{aff}S=\mathrm{span}S$.
• If $s_0\in \mathrm{aff}S$ then $\mathrm{aff}(S)-s_0=\mathrm{span}(S-s_0)=\mathrm{span}(S-S)$ is a linear subspace of $X$.
• $\mathrm{aff}(S-S)=\mathrm{span}(S-S)$ if $S\ne \varnothing$.
  • So, $\mathrm{aff}(S-S)$ is always a vector subspace of $X$ if $S\ne \varnothing$.
• If $S$ is convex then $\mathrm{aff}(S-S)=\bigcup _{\lambda >0}\lambda (S-S)$
• For every $s_0\in \mathrm{aff}S$, $\mathrm{aff}S=s_0+\mathrm{span}(S-s_0)=s_0+\mathrm{span}(S-S)=S+\mathrm{span}(S-S)=s_0+\mathrm{cone}(S-S)$ where $\mathrm{cone}(S-S)$ is the smallest cone containing $S-S$ (here, a set $C\subseteq X$ is a cone if $rc\in C$ for all $c\in C$ and all non-negative $r\ge 0$).
  • Hence $\mathrm{cone}(S-S)=\mathrm{span}(S-S)$ is always a linear subspace of $X$ parallel to $\mathrm{aff}S$ if $S\ne \varnothing$.
  • Note: $\mathrm{aff}S=s_0+\mathrm{span}(S-s_0)$ says that if we translate $S$ so that it contains the origin, take its span, and translate it back, we get $\mathrm{aff}S$. Moreover, $\mathrm{aff}S$ or $s_0+\mathrm{span}(S-s_0)$ is what $\mathrm{span}S$ would be if the origin was at $s_0$.

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 $\mathrm{cone}S$.
• 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 $\mathrm{span}S$ of $S$, which contains the affine hull of $S$.

Sources

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