Milman’s cuboid isoperimetric conjecture and Brezis Open Problem 10.1

Let $Q=(0,1)^3$ . For every measurable set $E\subset Q$ of finite perimeter and volume $|E|=v\in(0,1/2]$ , the relative perimeter $P(E;Q)$ satisfies

P(E;Q)\ge I_Q(v):= \begin{cases} \dfrac{\pi}{2}\left(\dfrac{6v}{\pi}\right)^{2/3},&0<v\le \dfrac{4\pi}{81},\\[6pt] \sqrt{\pi v},&\dfrac{4\pi}{81}\le v\le \dfrac{1}{\pi},\\[6pt] 1,&\dfrac{1}{\pi}\le v\le \dfrac12. \end{cases}

Moreover, equality cases, up to isometries of the cube and complementation, are as follows: for $0<v<4\pi/81$ , $E$ is a corner eighth-ball of radius $r=(6v/\pi)^{1/3}$ ; for $4\pi/81<v<1/\pi$ , $E$ is an edge quarter-cylinder of radius $r=2\sqrt{v/\pi}$ spanning a cube edge; and for $1/\pi<v\le1/2$ , $E$ is a coordinate slab of thickness $v$ . At the transition volumes $v=4\pi/81$ and $v=1/\pi$ , the corresponding adjacent families are also equality cases.

References

Primary source

arXiv

Additional references

Relative isoperimetry in the three-dimensional cube: classification and quantitative stability — arXiv — Bin Deng, Jiahuan Li, Yilu Liu

geometric-measure-theory isoperimetric-inequalities

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new unrefereed preprint claims to determine the three-dimensional cube profile and extend the result to a family of cuboids, but the claim has not been independently checked.

Milman’s cuboid isoperimetric conjecture proposes explicit minimizers for the relative perimeter at each prescribed volume, including corner pieces, edge pieces, and coordinate slabs. The related Brezis Open Problem concerns the profile near half volume; earlier work established important special cases but not the full three-dimensional statement.

Known results

Morgan: the conjectured minimizers hold for sufficiently small volumes, via balls near cube vertices.
Hadwiger, with a later Gaussian-contraction proof by Barthe and Maurey: the half-volume case.
Acerbi–Fusco–Morini: the profile is settled in a neighborhood of half volume.
Coordinate-polyhedral competitors: restricted minimization among sets bounded by coordinate-parallel hyperplanes is classified, but this does not settle the unrestricted problem.

October 2026 claimed classification

Bin Deng, Jiahuan Li, and Yilu Liu’s preprint Relative isoperimetry in the three-dimensional cube: classification and quantitative stability claims the complete three-dimensional profile, identifying corner eighth-balls, edge quarter-cylinders, and coordinate slabs, together with transition volumes and optimal quadratic stability. It also claims an extension to all cuboids in the stated one-parameter family. This is a complete-resolution claim from an unrefereed preprint and has no independent mathematical corroboration in the retrieved evidence.

Current status (as of October 2026): the full three-dimensional cuboid conjecture is claimed solved by the new preprint, while the claim remains unverified; the related half-volume result and other partial cases are established.

Sources

No solutions have been posted yet.