In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $\mathbb R^n$ as
$$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \ldots \theta_k = 1 \right\}.$$
Then, it claims $\text{aff } U = \mathbb R^2$ if $U$ is the unit circle. Why is this? Isn't any arc (or convex subset of the circle) entirely contained in the circle? I would think $\text{aff } U = U$ if $U$ is the unit circle.
Best Answer
I guess I know your confusion. The point is that $\theta_i$ is not constrained to be greater than zero. So now you may understand why three non-collinear points will fill the whole $\Bbb R^2$.