# Structure and stability for Brascamp–Lieb inequalities

Finner's Inequality is expressed using a set system $E$ of subsets of $\{1,\ldots,n\}$ where each set in the set system represents a function and the variables the function acts upon, and each set is associated with an exponent. The inequality bounds the integral of the product of the functions by the product of the $L^p$-norms of the functions where $p$ depends on the function and is equal to the associated exponent. Our project first looked to understand stability results from other researchers on simple set systems, such as $E = \{ \{1\}, \{1\} \}$, which gives Hölder's Inequality. Then, we generalized this result to show stability of the generalized $n$-function Hölder's Inequality, which has set system $E = \{ \{1\}, ... , \{1\} \}$, where $E$ contains $n$ singleton sets. When the inequality differed from equality by a constant of $\varepsilon$, we were able to bound the $L_2$ distance of the square roots of any pair of functions by $\varepsilon^{1/2}$. Then we showed stability of the set system $\{ \{1, 2\}, \{2, 3\}, \{3, 1\} \}$ with the same bound of $\varepsilon^{1/2}$. After finding an example using fully connected graphs that showed $1/2$ was the best possible exponent for this set system, we conjectured that this exponent would be the same for every set system. However, our proof techniques did not work the same on set systems of more variables. Instead, we were able to find an exponent that works that depends on $n$ and the exponents associated with the set systems, but were unable to show that this is the best exponent possible.