Convex Optimization - affine Set

A set $A$ is said to be an affine set if for any two distinct points, the line passing through these points lie in the set $A$.

Note −

$S$ is an affine set if and only if it contains every affine combination of its points.
Empty and singleton sets are both affine and convex set.

For example, solution of a linear equation is an affine set.

Proof

Let S be the solution of a linear equation.

By definition, $S=\left \{ x \in \mathbb{R}^n:Ax=b \right \}$

Let $x_1,x_2 \in S\Rightarrow Ax_1=b$ and $Ax_2=b$

To prove : $A\left [ \theta x_1+\left ( 1-\theta \right )x_2 \right ]=b, \forall \theta \in\left ( 0,1 \right )$

$A\left [ \theta x_1+\left ( 1-\theta \right )x_2 \right ]=\theta Ax_1+\left ( 1-\theta \right )Ax_2=\theta b+\left ( 1-\theta \right )b=b$

Thus S is an affine set.

If $C$ is an affine set and $x_0 \in C$, then the set $V= C-x_0=\left \{ x-x_0:x \in C \right \}$ is a subspace of C.

Let $x_1,x_2 \in V$

To show: $\alpha x_1+\beta x_2 \in V$ for some $\alpha,\beta$

Now, $x_1+x_0 \in C$ and $x_2+x_0 \in C$ by definition of V

Now, $\alpha x_1+\beta x_2+x_0=\alpha \left ( x_1+x_0 \right )+\beta \left ( x_2+x_0 \right )+\left ( 1-\alpha -\beta \right )x_0$

But $\alpha \left ( x_1+x_0 \right )+\beta \left ( x_2+x_0 \right )+\left ( 1-\alpha -\beta \right )x_0 \in C$ because C is an affine set.

Therefore, $\alpha x_1+\beta x_2 \in V$

Hence proved.