The Brascamp-Lieb inequality unifies and generalizes several of the most central inequalities in analysis, among others the inequalities of H\"older, Young and Loomis-Whitney. It has the form

$$\int_{H}\prod_{j=1}^{m} f_{j}^{p_{j}}(B_{j}x)\mathrm{d}x\leq C\prod_{j=1}^{m}\left(\int_{H_{j}}f_{j}\right)^{p_{j}} $$
where $H$ and $H_{j}$ are finite dimensional Hilbert spaces, $B_{j}: H\rightarrow H_{j}$ are linear maps, $p_{j}\geq 0$, $C<\infty$, $f_{j}\geq 0$. The plan of the talk is to discuss some natural questions that arise for a given set of linear maps $B_{j}$ as above:

- Under which conditions on $p_{j}$ is $C<\infty$ for all integrable $f_{j}$?
- Can one "plot'' the set of such $p_{j}$? Spoiler alert: It's a polytope.
- Can one compute the sharp constant of this inequality?
- Can one find $f_{j}$ for which equality holds?

No deep background in Analysis is needed. Linear algebra will do!