(Note: I asked this question on math stackexchange but did not get an answer. So I decide to ask this question here and hopefully somebody would know the answer/how to approach).

Consider the $d$-dimensional $\ell_1$-ball $\mathbb B_d := \{x: |x_1|+\cdots+|x_d|\leq 1\}$ and the $d$-dimensional $\ell_1$-surface $\mathbb S_d := \{x: |x_1|+\cdots+|x_d|=1\}$. I'm interested in the following volume ratio: $$ \mathrm{vol}(\mathbb B_d) / \mathrm{vol}(\mathbb S_{d-1}). $$

It is well-known that the volume ratio for $\ell_2$-balls and surfaces is $d$. It is also known that $\mathrm{vol}(\mathbb B_d) = 2^d/d!$. But it seems difficult to find $\mathrm{vol}(\mathbb S_d)$.