Relative isoperimetry in the three-dimensional cube: classification and quantitative stability

We determine the complete relative isoperimetric profile of the three-dimensional unit cube and classify all minimizing sets, including the equality cases at both transition volumes. Up to cube isometries and complementation, the minimizers are corner eighth-balls, edge quarter-cylinders, and coordinate slabs, with transitions at $4\pi \mathrm{/}81$ and $1\mathrm{/}\pi$. Building on this classification, we prove that the perimeter deficit at each fixed volume controls the squared $L^1$ distance to the full minimizing family, with optimal exponent two. Near either transition, a uniform estimate combines the distance to each adjacent phase with its excess perimeter and identifies the linear degeneration, at neighboring volumes, of the best constant for distance to the actual minimizers. As consequences, we resolve the isoperimetric conjecture in Milman’s formulation (Conjecture~1.1) for all cuboids $(0,\beta )\times (0,1{)}^2$, $0<\beta \le 1$, and the three-dimensional case of Brezis’s Open Problem~10.1.