We discuss a few conjectural Gaussian isoperimetric inequalities where traditional symmetrization or heat flow methods fail. Generally speaking, we are trying to maximize a convex function on a convex domain. So, any method that tries to maximize a function by moving in the direction of higher gradient should fail. Symmetrization and heat flow methods seem to fall into this category, so we need to develop new approaches to our problems.;The central object of study is the Ornstein-Uhlenbeck operator, i.e. the heat operator on Euclidean space equipped with the Gaussian measure. In this setting, Fourier analysis with respect to Hermite polynomials becomes most relevant. Due to the relation of the hypercube and the Gaussian measure via Central Limit Theorems, these Gaussian isoperimetric results imply inequalities for discrete functions, which then have applications to theoretical computer science.;The first problem we discuss asks for the maximum of the sum squared first-order Hermite-Fourier coefficients over all partitions of Euclidean space. The second problem we discuss asks for the most noise stable partitions of Euclidean space. The latter problem is a generalization of the so-called Gaussian double bubble problem. We give partial solutions to these two problems, and we also give some negative results for the latter problem.
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