Affine hull

In mathematics, the affine hull or affine span of a set S in Euclidean space Rn 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,

ExamplesEdit

  • 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 R3 is the entire space R3.

PropertiesEdit

For any subsets  

  •  
  •   is a closed set if   is finite dimensional.
  •  
  • If   then  .
  • If   then   is a linear subspace of  .
  •  .
    • So in particular,   is always a vector subspace of  .
  • If   is convex then  
  • For every  ,   where   is the smallest cone containing   (here, a set   is a cone if   for all   and all non-negative  ).
    • Hence   is always a linear subspace of   parallel to  .

Related setsEdit

  • If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all   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  , 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.

ReferencesEdit

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