On the relative isoperimetric problem for the cube
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915034266861568 |
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| author | Chambers, Gregory R. Mouillé, Lawrence |
| author_facet | Chambers, Gregory R. Mouillé, Lawrence |
| contents | 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 $[0,ε]^3$, $[0,ε]^2 \times [0,1]$, or $[0,ε] \times [0,1]^2$ for some $ε> 0$.
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 $\left( B^3(ε) \right) \cap [0,1]^3$, $\left( B^2(ε) \times [0,1] \right) \cap [0,1]^3$, or $[0,ε] \times [0,1]^2$ for some $ε> 0$. Here, $B^k(ε)$ is the closed ball in $\mathbb{R}^k$ of radius $ε$ centered at the origin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_04382 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the relative isoperimetric problem for the cube Chambers, Gregory R. Mouillé, Lawrence Differential Geometry 49Q10 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 $[0,ε]^3$, $[0,ε]^2 \times [0,1]$, or $[0,ε] \times [0,1]^2$ for some $ε> 0$. 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 $\left( B^3(ε) \right) \cap [0,1]^3$, $\left( B^2(ε) \times [0,1] \right) \cap [0,1]^3$, or $[0,ε] \times [0,1]^2$ for some $ε> 0$. Here, $B^k(ε)$ is the closed ball in $\mathbb{R}^k$ of radius $ε$ centered at the origin. |
| title | On the relative isoperimetric problem for the cube |
| topic | Differential Geometry 49Q10 |
| url | https://arxiv.org/abs/2302.04382 |