Affine hull

In mathematics, the affine hull of a set in Euclidean space is the smallest affine set containing , or equivalently, the intersection of all affine sets containing . Here, an affine set may be defined as the translation of a vector subspace.

The affine hull of is the set of all affine combinations of elements of , that is,

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 is the entire space .

Properties

  • is a closed set
  • 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.

References

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