When I was reading 'Convex Optimization', Stephan Boyd, I was stopped by Example 2.2. Before Example 2.2 is started, following definition is coming.
- If the affine dimension of a set $C\subseteq\mathbf{R}^n$ is less than $n$, then the set lies in the affine set $\mathbf{aff}\;C\neq$. We define the $\;relative\;interior$ of the set $C$, as its interior relative to $\mathbf{aff}\;C:$ $$\text{relint}\;C=\{x\in C\,|\,B(x,r)\cap \text{aff}\;C \subseteq C\;\text{for some}\; r>0\},$$ where $B(x,r)=\{y\,|\,\|y-x\|\le r\}$, the ball of radius $r$ and center $x$ in the norm $\|\bullet\|$. (Here $\|\bullet\|$ is any norm; all norms define the same relative interior.)
- $\mathbf{Example\;2.2}$ Consider a square in the $(x_1,x_2)$-plane in $\mathbf{R}^3$, defined as $$C=\{x\in\mathbf{R}^3\,|\,-1\le x_1\le 1,\, -1\le x_2\le 1,\, x_3=0\}$$ Its affine hull is the $(x_1,x_2)$-plane, $i.e.,\;\text{aff}\;C=\{x\in\mathbf{R}^3\;|\;x_3=0\}$. The interior of $C$ is empty, but the relative interior is
$$\text{relint}\;C=\{x\in\mathbf{R}^3\,|\,-1\lt x_1\lt 1,\, -1\lt x_2\lt 1,\, x_3=0\}$$
I wondering why all norms define the same relative interior. And how we know relint $C$ in $\mathbf{Example\;2.2}$ which is not determined any ball's information(center $x$ and radius $r$)
