Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry.
Let $\partial C$ denote the boundary of $C$. My question is :
Can we find a norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere ? Moreover, is it unique?
This is question arises after reading my course, but I can not find a satisfactory 'solution'.
$C$ is a convex and balanced (symmetric) neighbourhood of $0$. Since it's a neighbourhood of $0$, it is absorbing, hence the Minkowski functional
$$\mu_C \colon x \mapsto \inf \left\{ t > 0 : x \in t\cdot C\right\}$$
of $C$ is finite everywhere. Since $C$ is convex, it is a sub-norm (subadditive, positive homogeneous, and non-negative), and since $C$ is balanced, it is a seminorm. Since $C$ is bounded, $\mu_C$ is a norm.
The open unit ball of $\mu_C$ is the interior of $C$, and the closed unit ball of $\mu_C$ is the closure of $C$, hence the unit sphere of $\mu_C$ is the boundary $\partial C$ of $C$.
The uniqueness follows since a norm is determined by its unit sphere.
It may be worth pointing out that nothing in the argument is specific to $\mathbb{R}^2$ or even the finite-dimensionality. For any convex, balanced and bounded neighbourhood $C$ of $0$ in a normed space $E$, the Minkowski functional $\mu_C$ is a norm that is equivalent to the norm we started with ($\mu_C$ is continuous since $C$ is a neighbourhood of $0$, and the topology induced by $\mu_C$ is at least as fine as the original since $C$ is bounded), whose unit sphere is $\partial C$.