In the textbook "Convex Optimization", S. Boyd says that the affine hull of a set $C\subseteq \mathbb{R}^{^{n}}$ is the smallest affine set that contains C.
Moreover, the Ex. 2.2 shows the set $ C=\left \{ x\in \mathbb{R}^{3} |-1\leq x_{1},x_{2}\leq 1,x_{3}=0\right \}$ has the affine set aff$ C =\left \{ x\in \mathbb{R}^{3} |x_{3}=0\right \}$
In my opinion, i think that aff C defined above is not tight because it is not the smallest affine set. What do you think about it?
On
Old thread but still nice to have it clarified. The definition "smallest affine set" doesn't necessarily mean this is a closed segment (in $\Bbb R^2$ for instance), because the affine combination doesn't have $\theta_i \ge 0$ restriction. So in this case the smallest affine set is actually indeed the $x_1,x_2$ plane.
The book is right, it is indeed the smallest: it needs to contain the square on the $x_1,x_2$ plane, hence is has to contain at least that plane.
:)