Department of MathematicsProject leader: Professor Nicholas Cook
Project manager: Haotian Gu
Team members: Harry Chen, Jean-Luc Rabideau, Alden Wang
Finner's inequality is a large family of inequalities with many applications in different branches of mathematics. It relates the integrals of products of functions with the product of integrals of functions. Some special cases of Finner's inequality include the Cauchy-Schwarz Inequality, which relates the length of two vectors with their dot product. Another application is the Loomis-Whitney Inequality, which relates the volume of an object with the areas of its shadows. The types of functions where equality holds in Finner's Inequality are already known. Our project instead looked at when equality is close to holding. If there is almost equality in the expression proposed by Finner, then are the functions in question necessarily close to these known types of functions where equality does hold? This is what it means for an inequality to be stable. Our project investigated the stability of Finner's inequality.
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.