Cookie cutters: Bisections with fixed shapes
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915170040676352 |
|---|---|
| author | Schnider, Patrick Soberón, Pablo |
| author_facet | Schnider, Patrick Soberón, Pablo |
| contents | In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $\mathbb{R}^d$. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set $K$. We distinguish the problem when we are allowed to use scaled and translated copies of $K$ and the problem when we are allowed to use scaled isometric copies of $K$. These problems have only previously been studied if $K$ is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any $d+1$ masses for star-shaped compact sets $K$ with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of $K$. Additional proofs are included for particular instances of $K$, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cookie cutters: Bisections with fixed shapes Schnider, Patrick Soberón, Pablo Combinatorics In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $\mathbb{R}^d$. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set $K$. We distinguish the problem when we are allowed to use scaled and translated copies of $K$ and the problem when we are allowed to use scaled isometric copies of $K$. These problems have only previously been studied if $K$ is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any $d+1$ masses for star-shaped compact sets $K$ with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of $K$. Additional proofs are included for particular instances of $K$, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems. |
| title | Cookie cutters: Bisections with fixed shapes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.17176 |