Isoperimetric inequalities on slabs with applications to cubes and Gaussian slabs

We study isoperimetric inequalities on “slabs”, namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension‐one base. As our two main applications, we consider the case when the base is the flat torus and the standard Gaussian measure on . The isoperimetric conjecture on the three‐dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to 0, and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three‐dimensional cube with side lengths in a new range of relative volumes . In particular, we confirm the conjecture for the standard cube () for all , when for the entire range where spheres are conjectured to be minimizing, and also for all . When we reduce the validity of the full conjecture to establishing that the half‐plane is an isoperimetric minimizer. We also show that the analogous conjecture on a high‐dimensional cube is false for . In the case of a slab with a Gaussian base of width , we identify a phase transition when and when . In particular, while products of half‐planes with are always minimizing when , when they are never minimizing, being beaten by Gaussian unduloids. In the range , a potential trichotomy occurs.