I tried to do the following exercise:
Show that the extension given in the Hahn-Banach theorem is not always unique.
My example was $g: A \subset \mathbb{R}^2 \to \mathbb{R}$ given by $g(x_1, 0) = x_1$, where $A$ is the $x$-axis of $\mathbb{R}^2$, with $p: \mathbb{R}^2 \to \mathbb{R}$ given by $p(a, b) = |a| + 3|b|$. Clearly $g(x) \leq p(x)$ for all $x \in A$ and $p$ is sublinear. And the functions $f_1 : \mathbb{R}^2 \to \mathbb{R}$ and $f_2 : \mathbb{R}^2 \to \mathbb{R}$ defined by $f_1(a, b) = a + b$ and $f_2(a, b) = a + 2b$ are both linear, extend $g$ and satisfy $f_1(x) \leq p(x)$, $f_2(x) \leq p(x)$ for any $x \in \mathbb{R}^2$.
The reason I'm asking this question is because I looked up other examples in this site and all the ones I found were much more complicated. Is there anything wrong with my example?
EDIT: I realized things might be a bit subtler than this. I don't know what the Hahn-Banach extension of $g$ will be... but I know it's linear so it's of the form $f(a, b) = a + b f(e_2)$. So let $p$ be the sublinear function given by $p(a, b) = |a| + c|b|$ where $c = |f(e_2)| + 1$, a constant. Now, $f$ is dominated by $p$ and another extension could be $f_2(a, b) = a + \frac{c}{2} b$. Is that correct or was the original thing just fine?
Your first two examples are fine. But you are making some circular arguments the the EDIT. It doesn't make sense to make $p$ dependent on the extension $f$.