Hessian of Bellman functions and uniqueness of Brascamp--Lieb inequality

Under some assumptions on the vectors $a_1,\mathrm{.}\mathrm{.},a_n\in {\mathbb{R}}^k$ and the function $B:{\mathbb{R}}^n\to \mathbb{R}$ we find the sharp estimate of the expression $\int_{$\mathbb{R}$^{k}} B(u_{1}(a_{1}\cdot x),…, u_{n}(a_{n}\cdot x))dx$intermsof$\int_{$\mathbb{R}$}u_{j}(y)dy, j=1,…,n.$Insomeparticularcasewewillshowthattheseassumptionson$B$ imply that there is only one Brascamp–Lieb inequality.