Let $K$ is centrally symmetric convex body containing the $\ell_2$ unit sphere $B_n^2$. The polar body of $K$ is define as $$ K^{\circ} = \left\lbrace y\in \mathbb{R}^{n} \mid \left< x,y \right> \le 1, \forall x\in K \right\rbrace $$
I'm trying to figure out the following question:
Let $C$ be the contact points of $K$ with $B_n^2$ (and assume it is not empty). Show that $C \subseteq K^\circ$.
I know that $\left(K^{\circ}\right)^{\circ} = K$ and that $\left(B_n^2\right)^{\circ} = B_n^2$. I also know that for $y\in C$ we have $\left\Vert y \right\Vert_{2} = \left\Vert y \right\Vert_{K} = 1$. The above question is equivalent to showing $\left\Vert y \right\Vert_{K^{\circ}} = 1$.
I fail to see why given $y\in C$, $\forall x \in K, \left<x, y\right> \le 1$. Any (subtle) hints?
Let $y$ be a contact point between $K$ and $B$. That means $y$ is on the boundary of $K$ and also on the boundary of $B$. Let $H$ a supporting hyperplane for $K$ at $y$. Since $K\supset B$, $H$ will be a supporting hyperplane for $B$ at $y$. But $B$ has a unique supporting hyperplane at $y$, given by $H=\{x\ | \ \langle y,x\rangle= 1\}$. We conclude $K\subset \{x\ | \ \langle y,x\rangle\le 1\}$.