Isoperimetric inequalities on slabs with applications to cubes and Gaussian slabs

Abstract 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.