Affine hull

Examples

• The affine hull of the empty set is the empty set.
• The affine hull of a singleton 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.

  Properties

• 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 .
• * Hence is always a linear subspace of parallel to.

  Related sets

• 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.