Affine hull

 ( Affine Geometry ) 

In mathematics, the affine hull or affine span of a set $S$ in Euclidean space $\backslash mathbb\{R\}^n$ 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 $\backslash operatorname\{span\}\; S$ 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,

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

• 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 $\backslash mathbb\{R\}^3$ is the entire space $\backslash mathbb\{R\}^3$.

• $\backslash operatorname\{aff\}(\backslash operatorname\{aff\}\; S)\; =\; \backslash operatorname\{aff\}\; S\; \backslash subset\; \backslash operatorname\{span\}\; S\; =\; \backslash operatorname\{span\}\; \backslash operatorname\{aff\}\; S$.
• $\backslash operatorname\{aff\}\; S$ is a closed set if $X$ is finite dimensional.
• $\backslash operatorname\{aff\}(S\; +\; T)=\backslash operatorname\{aff\}\; S\; +\; \backslash operatorname\{aff\}\; T$.
• $S\backslash subset\; \backslash operatorname\{aff\}\; S$.
• If $0\; \backslash in\; \backslash operatorname\{aff\}\; S$ then $\backslash operatorname\{aff\}\; S\; =\; \backslash operatorname\{span\}\; S$.
• If $s\_0\; \backslash in\; \backslash operatorname\{aff\}\; S$ then $\backslash operatorname\{aff\}(S)\; -\; s\_0\; =\; \backslash operatorname\{span\}(S\; -\; s\_0)=\; \backslash operatorname\{span\}(S\; -\; S)$ is a linear subspace of $X$.
• $\backslash operatorname\{aff\}(S\; -\; S)\; =\; \backslash operatorname\{span\}(S\; -\; S)$ if $S\backslash ne\backslash varnothing$.
• So, $\backslash operatorname\{aff\}(S\; -\; S)$ is always a vector subspace of $X$ if $S\backslash ne\backslash varnothing$.
• If $S$ is convex set then $\backslash operatorname\{aff\}(S\; -\; S)\; =\; \backslash displaystyle\backslash bigcup\_\{\backslash lambda\; >\; 0\}\; \backslash lambda\; (S\; -\; S)$
• For every $s\_0\; \backslash in\; \backslash operatorname\{aff\}\; S$, $\backslash operatorname\{aff\}\; S\; =\; s\_0\; +\; \backslash operatorname\{span\}(S\; -\; s\_0)\; =\; s\_0\; +\; \backslash operatorname\{span\}(S\; -\; S)\; =\; S\; +\; \backslash operatorname\{span\}(S\; -\; S)\; =\; s\_0\; +\; \backslash operatorname\{cone\}(S\; -\; S)$ where $\backslash operatorname\{cone\}(S\; -\; S)$ is the smallest Convex cone containing $S\; -\; S$ (here, a set $C\; \backslash subseteq\; X$ is a cone if $r\; c\; \backslash in\; C$ for all $c\; \backslash in\; C$ and all non-negative $r\; \backslash geq\; 0$).
• Hence $\backslash operatorname\{cone\}(S\; -\; S)=\; \backslash operatorname\{span\}(S\; -\; S)$ is always a linear subspace of $X$ parallel to $\backslash operatorname\{aff\}\; S$ if $S\backslash ne\backslash varnothing$.
• Note: $\backslash operatorname\{aff\}\; S\; =\; s\_0\; +\; \backslash operatorname\{span\}(S\; -\; s\_0)$ says that if we translate $S$ so that it contains the origin, take its span, and translate it back, we get $\backslash operatorname\{aff\}\; S$. Moreover, $\backslash operatorname\{aff\}\; S$ or $s\_0\; +\; \backslash operatorname\{span\}(S\; -\; s\_0)$ is what $\backslash operatorname\{span\}\; S$ would be if the origin was at $s\_0$.

• If instead of an affine combination one uses a convex combination, that is, one requires in the formula above that all $\backslash alpha\_i$ 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 $\backslash operatorname\{cone\}\; S$.
• If however one puts no restrictions at all on the numbers $\backslash alpha\_i$, instead of an affine combination one has a linear combination, and the resulting set is the linear span $\backslash operatorname\{span\}\; S$ of $S$, which contains the affine hull of $S$.

• R.J. Webster, Convexity, Oxford University Press, 1994. .