On the relative isoperimetric problem for the cube

In this article, we solve the relative isoperimetric problem in $[0,1{]}^3$ for orthogonal polyhedra. Up to isometries of the cube or sets of measure $0$, the minimizers are of the form $[0,\epsilon {]}^3$, $[0,\epsilon {]}^2\times [0,1]$, or $[0,\epsilon ]\times [0,1{]}^2$ for some $\epsilon >0$. This should be compared to the conjectured minimizers for the unconstrained relative isoperimetric problem in $[0,1{]}^3$, which are (up to isometries and sets of measure $0$) of the form $(B^3(\epsilon ))\cap [0,1{]}^3$, $(B^2(\epsilon )\times [0,1])\cap [0,1{]}^3$, or $[0,\epsilon ]\times [0,1{]}^2$ for some $\epsilon >0$. Here, $B^k(\epsilon )$ is the closed ball in ${\mathbb{R}}^k$ of radius $\epsilon$ centered at the origin.