On the relative isoperimetric problem for the cube

Abstract

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 $[0,\hat{I}\mu {]}^3$, $[0,\hat{I}\mu {]}^2\times [0,1]$, or $[0,\hat{I}\mu ]\times [0,1{]}^2$ for some $\hat{I}\mu >0$. 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 $\left(B^3(\hat{I}\mu )\right)\cap [0,1{]}^3$, $\left(B^2(\hat{I}\mu )\times [0,1]\right)\cap [0,1{]}^3$, or $[0,\hat{I}\mu ]\times [0,1{]}^2$ for some $\hat{I}\mu >0$. Here, $B^k(\hat{I}\mu )$ is the closed ball in ${\mathbb{R}}^k$ of radius $\hat{I}\mu$ centered at the origin.

19 pages, 3 figures, comments welcome