Let $\phi$ be 1-homogeneous, convex and coercive from $R^n$ to $[0,\infty]$.
Define the Wulff Shape associated to $\phi$ as $$ W_{\phi}=\cap_{y\in S^{n-1}}\{ x : x\cdot y<\phi(y) \}. $$
My question is about the following fact:
Let $A$ be an open, bounded and convex set containing the origin, then $A=W_{\phi}$ where $\phi(x)=\sup\{x\cdot y: y\in A\}. $
Can someone explain me how to prove it? I am trying with no success.
I was able to prove only one inclusion:
Let $x\in A$. If $y\in S^{n-1}$, then clearly $x\cdot y\leq \phi(y)$ (since $x\in A$ and by definition of $\phi$), moreover the equality can not occur since $\phi(y)=z\cdot y$ for a certain $z\in\partial A$ and $x\in A$ which is an open set.