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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.08707 |
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| _version_ | 1866912755701776384 |
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| author | Bludov, Mikhail V. |
| author_facet | Bludov, Mikhail V. |
| contents | For a finite set of points $V=\{v_1, \dots, v_m\}$ in Euclidean space $\mathbb{R}^d$ and a point $r \in \mathbb{R}^d$, a subset $S \subset V$ is called $r$-balanced if $\mathrm{relint}(\mathrm{conv}(S)) \cap r \neq \emptyset$. In the case when $r$ is a point in the relative interior of the whole set $\mathrm{conv}(V)$, we prove that the poset of all balanced subsets, excluding the whole set $V$, is homotopy equivalent to the sphere of dimension $m-k-2$, where $k$ is the dimension of the affine hull of $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Homotopy Type of Balanced subsets Bludov, Mikhail V. Combinatorics For a finite set of points $V=\{v_1, \dots, v_m\}$ in Euclidean space $\mathbb{R}^d$ and a point $r \in \mathbb{R}^d$, a subset $S \subset V$ is called $r$-balanced if $\mathrm{relint}(\mathrm{conv}(S)) \cap r \neq \emptyset$. In the case when $r$ is a point in the relative interior of the whole set $\mathrm{conv}(V)$, we prove that the poset of all balanced subsets, excluding the whole set $V$, is homotopy equivalent to the sphere of dimension $m-k-2$, where $k$ is the dimension of the affine hull of $V$. |
| title | On the Homotopy Type of Balanced subsets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.08707 |