The question is basically that, since I heard that the Loomis-Whitney inequality is a special case of the Brascamp-Lieb inequality, I would like to check the constant factor in B-L inequality is indeed $1$ as in L-W inequality.
So in this case we have $n=m=d$, $\forall i\in[d],n_i=d-1,c_i=\frac{1}{d-1}$, and $B_i(x_1,\cdots,x_d)=(x_1,\cdots,x_{i-1},x_{i+1},\cdots,x_d)$. The constant factor being $1$ means
$$1=\sup\left\{\frac{\prod\limits_{i=1}^d(\mathrm{det}(A_i))^{1/(d-1)}}{\mathrm{det}(\frac{1}{d-1}\sum\limits_{i=1}^d B_i^*A_iB_i)}: \quad A_i\in\mathcal{M}_{(d-1)\times(d-1)},A_i>0,\forall i\in[d]\right\}$$
where the supremum takes over all $(d-1)$-dimensional positive-definite matrices.
My intuition is to take logarithm and try to use the log-concavity of determinant, but I literally don't know where to start to deal with the sum $\sum\limits_{i=1}^d B_i^*A_iB_i$. Every term in the sum is obtained from $A_i$ by adding a zero-column and a zero-row after the $(i-1)$-th column and the $(i-1)$-row respectively, then how do we study their sum?