Loomis-Whitney inequalities on corank $1$ Carnot groups

In this paper we provide another way to deduce the Loomis-Whitney inequality on higher dimensional Heisenberg groups $\mathbb{H}^n$ based on the one on the first Heisenberg group $\mathbb{H}^1$ and the known nonlinear Loomis-Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank $1$ Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp--Lieb inequality and the subadditivity of the entropy developed in arXiv:0710.0870v2.