Inscribed Tverberg-Type Partitions for Orbit Polytopes
| Autor: | Simon, Steven, Timofeyev, Tobias |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Mathematika 68 (2022) 1135-1152 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/mtk.12160 |
| Popis: | Tverberg's theorem states that any set of $t(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Moreover, generic collections of fewer points cannot be so divided. Extending earlier work of the first author, we show that one can nonetheless guarantee inscribed ``polytopal partitions" with specified symmetry conditions in many such circumstances. Namely, for any faithful and full--dimensional orthogonal representation $\rho\colon G\rightarrow O(d)$ of any order $r$ group $G$, we show that a generic set of $t(r,d)-d$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets so that there are $r$ points, one from each of the resulting convex hulls, which are the vertices of a convex $d$--polytope whose isometry group contains $G$ via the regular action afforded by the representation. As with Tverberg's theorem, the number of points is optimal for this. At one extreme, this gives polytopal partitions for all regular $r$--gons in the plane, as well as for three of the six regular 4--polytopes in $\mathbb{R}^4$. At the other extreme, one has polytopal partitions for $d$-polytopes on $r$ vertices with isometry group equal to $G$ whenever $G$ is the isometry group of a vertex--transitive $d$-polytope. Comment: 17 pages; 1 figure |
| Databáze: | arXiv |
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