Affine sets and affine hull

Question

Mathematically an affine hull can be expressed as

$ Aff[C] = \{\theta_1x_1 + \theta_2x_2 .... \theta_nx_n| x_i \in C \ \ \sum_{i=1}^{n}\theta_i = 1 \}$

Intuitively can anyone explain what this means?

Also, what is a 'hull'?

Answer 1

A little work shows that if $a,b \in \operatorname{aff} C$, then for all $\lambda \in \mathbb{R}$, we have $\lambda a + (1-\lambda) b \in \operatorname{aff} C$. In fact, this is one definition of an affine set. You can think of it as relaxation of convexity (for a convex set, $\lambda \in [0,1]$).

Suppose $c \in C$. Let $L = \{ a -c | a \in \operatorname{aff} C \}$.

Suppose $x \in L$, then $x+c \in \operatorname{aff} C$. Then $\lambda (x+c) + (1-\lambda) c = \lambda x +c \in \operatorname{aff} C$ for all $\lambda$, and so $\lambda x \in L$ for all $\lambda$.

Suppose $x,y \in L$, then $x+c, y+c \in \operatorname{aff} C$, and so $\frac{1}{2}((x+c)+(y+c)) = \frac{1}{2}(x+y) +c \in \operatorname{aff} C$. Hence $\frac{1}{2}(x+y) \in L$, and the previous result shows that $2 \frac{1}{2}(x+y) = x+y \in L$.

This shows that $L$ is a subspace.

Hence we can write $\operatorname{aff} C = L + \{c\}$, so $\operatorname{aff} C$ is basically a translate of a linear subspace.

The set $\operatorname{aff} C$ is the smallest affine set containing $C$, so it is the smallest affine 'container' of $C$. I presume this is the origin of the term 'hull', much as the hull of a ship contains the 'stuff' inside the ship.

The term linear hull is used for smallest subspace containing a set, and the term convex hull is used for smallest convex set containing a set.