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?
Best Answer
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.
:)