Let $X$ consist of two points $a$, $b$, put $\mu(\{a\})=\mu(\{b\})=1/2$, and let $L^p(\mu)$ be he resulting real $L^p-$space. Identify each real function $f$ on $X$ with the point $(f(a),f(b))$ in the plane.
For $0<p\leq \infty$, which are the units balls of $L^p(\mu)$?
For which $p$ is this unit ball a square? And a circle? How would the situation change if $\mu(\{a\})\not = \mu(\{b\})$?
This is Problem 1 in Chapter 5 of Rudin's 'Real and Complex Analysis'.
I think that the unit balls would be simply the following sets: $$\{f : (|f(a)|^p+|f(b)|^p)^{1/p}=1\},$$ Hence, this unit ball is a square for $p=\infty$ and for $p=1$ (in this case the square is rotated $\pi/4$), and it is a circle for $p=2$. Is that correct?
I don't know how the situation would change if the unitary sets didn't have the same measure.
Thanks in advance!