Dual norm of integral vector norm

Question

For $x\in\mathbb{C}^n$, consider the following norm:

$\|x\|_{B} := \int_{B}|\langle x,y\rangle| \ d\mu(y),$

where $B:=\{y\in\mathbb{C}^n \ | \ \|y\|_2 \leq 1\}$ is the euclidean-norm ($\|\circ\|_2$) unit ball.

My question is: what is the dual norm of $\|\circ\|_{B}$?

In other words, is there a ''nice'' (alternative) closed form expression for

$\|z\|_{B}^* = \sup_{\{x \ | \ \|x\|_{B}\leq 1\}}\langle z,x\rangle$ ?

I understand that this is a vague question since ''nice'' is subjective and the expression above is technically a closed form expression. However, any alternative expression or intuition on how to derive an equivalent expression would be much appreciated. Any related definitions in the literature would also help.

Answer 1

If $\mu$ is the Lebesgue measure. By substitution it is clear that,

$$\left\|x\right\|_B = \left\|x\right\|_2\int_{y\in B} \left|y_n\right| \mathrm d y$$ and this proves that, $$\left\|z\right\|_B^* = \frac1{\displaystyle \int_{y\in B} \left|y_n\right| \mathrm d y} \left\|z\right\|_2$$

Answer 2

If $\mu$ is invariant by the orthogonal group, then $\|x\|_B$ is proportional to the Euclidean norm,and up to a multiplicative constant the norm is self dual.