On the relative isoperimetric problem for the cube

TL;DR

This paper solves the relative isoperimetric problem within the cube for orthogonal polyhedra, identifying the minimal boundary sets as certain rectangular prisms, and compares these to conjectured minimizers involving spherical caps.

Contribution

It provides a complete solution to the relative isoperimetric problem for orthogonal polyhedra in the cube, characterizing the minimizers explicitly.

Findings

Minimizers are rectangular prisms of specific forms.

Comparison with conjectured spherical minimizers.

Results are up to isometries and measure-zero sets.

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 $[0,\epsilon]^3$, $[0,\epsilon]^2 \times [0,1]$, 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 $\left( B^3(\epsilon) \right) \cap [0,1]^3$, $\left( B^2(\epsilon) \times [0,1] \right) \cap [0,1]^3$, 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.