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arXiv · 2610.18351

An explicit dimension-free isoperimetric plateau in the cube

Lei Yu
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Abstract

We prove that, in every dimension, the relative Euclidean isoperimetric profile of the unit cube equals one whenever the prescribed volume differs from one half by at most $6.25\times10^{-12}$. The only minimizers in this interval are coordinate slabs. This answers Glaudo's question on the existence of a dimension-free central plateau. The proof combines a dimension-free local optimality estimate for slabs, established through BV slicing and a small-support estimate, with a quantitative Gaussian rigidity argument that forces the normal of an approximating half-space to be close to a coordinate direction.

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Lei Yu. 2026-10-13. An explicit dimension-free isoperimetric plateau in the cube. https://arxiv.org/abs/2610.18351

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