We will first survey some variational proofs of the Euclidean and Gaussian isoperimetric inequalities. One benefit of these proofs is they identify the local minima as opposed to just global minima. We then discuss two ways to generalize these inequalities. First, we can replace one set with multiple sets. Second, we can replace the surface area by the heat content, which is known as noise stability in the Gaussian setting. Finally, we will end by stating some open problems.
Partly based on joint work with Alex Tarter.
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