The affine hull and convex hull are closely related concepts. Let be a set in . The affine hull of is the set of all affine combinations of elements of :
The convex hull of is the set of all convex combinations of the elements of :
Putting the definitions side by side, we see that the only difference is that for the convex hull, the weights in the linear combination (the ‘s) have an additional restriction of being non-negative. The only restriction on the ‘s for combinations in the affine hull is that they sum to 1. This also means that the affine hull always contains the convex hull.
Let’s have a look at a few examples. If consists of two points, the convex hull is the line segment joining them (including the endpoints) while the affine hull is the entire line through these two points:
For 3 non-collinear points in two dimensions, the convex hull is the triangle with these 3 points as vertices while the affine hull is the entire plane :
The final illustration below shows the affine and convex hulls of 3 non-collinear points in . They are both two-dimensional sets living in 3-dimensional space. Notice also how the affine hull need not pass through the origin (whereas all subspaces of must pass through it).