Affine hull

Inmathematics,theaffine hulloraffine spanof asetSinEuclidean spaceRⁿis the smallestaffine setcontainingS,or equivalently, theintersectionof all affine sets containingS.Here, anaffine setmay be defined as thetranslationof avector subspace.

The affine hull aff(S) ofSis the set of allaffine combinationsof elements ofS,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 theempty setis the empty set.
• The affine hull of asingleton(a set made of one single element) is the singleton itself.
• The affine hull of a set of two different points is thelinethrough them.
• The affine hull of a set of three points not on one line is theplanegoing through them.
• The affine hull of a set of four points not in a plane inR³is the entire spaceR³.

Properties

For any subsets$S,T\subseteq X$

• $\operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S$
• $\operatorname {aff} S$is aclosed setif$X$is finite dimensional.
• $\operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T$
• If$0\in S$then$\operatorname {aff} S=\operatorname {span} S$.
• If$s_{0}\in S$then$\operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})$is a linear subspace of$X$.
• $\operatorname {aff} (S-S)=\operatorname {span} (S-S)$.
  • So in particular,$\operatorname {aff} (S-S)$is always a vector subspace of$X$.
• If$S$isconvexthen$\operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)$
• For every$s_{0}\in S$,$\operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)$where$\operatorname {cone} (S-S)$is the smallestconecontaining$S-S$(here, a set$C\subseteq X$is aconeif$rc\in C$for all$c\in C$and all non-negative$r\geq 0$).
  • Hence$\operatorname {cone} (S-S)$is always a linear subspace of$X$parallel to$\operatorname {aff} S$.

Related sets

• If instead of an affine combination one uses aconvex combination,that is, one requires in the formula above that all$\alpha _{i}$be non-negative, one obtains theconvex hullofS,which cannot be larger than the affine hull ofS,as more restrictions are involved.
• The notion ofconical combinationgives rise to the notion of theconical hull
• If however one puts no restrictions at all on the numbers$\alpha _{i}$,instead of an affine combination one has alinear combination,and the resulting set is thelinear spanofS,which contains the affine hull ofS.

Sources

• R.J. Webster,Convexity,Oxford University Press, 1994.ISBN0-19-853147-8.
• Roman, Stephen(2008),Advanced Linear Algebra,Graduate Texts in Mathematics(Third ed.), Springer,ISBN978-0-387-72828-5