Let X be a real normed space, $f \in X*, \mid\mid f\mid\mid=1 \text{ and let }\sigma \in (0,1)$.

$C:=\{ x\in X: f(x)\ge \sigma\mid\mid x\mid\mid \}$.

Prove that:

$C^{\circ}=\{ x\in X\colon f(x)> \sigma \mid\mid x \mid \mid \} $.

Can you help me with this one?