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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?

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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.

:)

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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.

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