TITLE: Why is the affine hull of the unit circle $\mathbb R^2$?
QUESTION [20 upvotes]: 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.

REPLY [3 votes]: Take any point in $\mathbb R$. We can always draw a line through it which passes through two points in the circle. That means it lies on the same line that passes through those points on the unit circle.
That's the definition of Affine Hull. Please rectify me if I'm wrong.