AI 中文总结
本文完整分类三维立方体中的相对等周极小化集合，证明周长亏缺控制平方L1距离，并解决相关猜想与开放问题。
AI 中文摘要
我们确定了三维单位立方体的完整相对等周剖面，并对所有极小化集合进行了分类，包括两个过渡体积处的等式情形。在立方体等距变换和补集意义下，极小化集合为角部八分之一球、棱边四分之一圆柱和坐标切片，过渡点位于$4\pi/81$和$1/\pi$。基于此分类，我们证明在每个固定体积下，周长亏缺控制到完全极小化族的平方$L^1$距离，且指数为最优的2。在任一过渡点附近，一个一致估计将到每个相邻相的距离与其多余周长相结合，并识别出在相邻体积下，到实际极小化集合距离的最佳常数的线性退化。作为推论，我们解决了Milman表述中的等周猜想（猜想1.1）对所有长方体$(0,\beta)\times(0,1)^2$（$0<\beta\leq1$）的情形，以及Brezis开放问题10.1的三维情形。
英文摘要
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 $4π/81$ and $1/π$. 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 $(0,β)\times(0,1)^2$, $0<β\leq1$, and the three-dimensional case of Brezis's Open Problem~10.1.
