The hyperspace of k-dimensional closed convex sets
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929522766512128 |
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| author | Escobedo-Bustamante, Adriana Jonard-Pérez, Natalia |
| author_facet | Escobedo-Bustamante, Adriana Jonard-Pérez, Natalia |
| contents | For every $n \geq 2$, let $K_k^n$ denote the hyperspace of all $k$-dimensional closed convex subsets of the Euclidean space $R^n$ endowed with the Atouch-Wets topology. Let $ K_{k,b}^n$ be the subset of $K_k^n$ consisting of all $k$-dimensional compact convex subsets. In this paper we explore the topology of $K_k^n$ and $K_{k,b}^n$ and the relation of these hyperspaces with the Grassmann manifold $G_k(n)$. We prove that both $K_k^n$ and $K_{k,b}^n$ are Hilbert cube manifolds with a fiber bundle structure over $G_k(n)$. We also show that the fiber of $K_{k,b}^n$ with respect to this fiber bundle structure is homeomorphic with $\mathbb R^{\frac{k(k+1)+2n}{2}}\times Q$, where $Q$ stands for the Hilbert cube. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The hyperspace of k-dimensional closed convex sets Escobedo-Bustamante, Adriana Jonard-Pérez, Natalia General Topology Geometric Topology Primary: 52A20, 52A21, 54B20, 54C10, 54C55. Secondary: 54H15, 57S25 For every $n \geq 2$, let $K_k^n$ denote the hyperspace of all $k$-dimensional closed convex subsets of the Euclidean space $R^n$ endowed with the Atouch-Wets topology. Let $ K_{k,b}^n$ be the subset of $K_k^n$ consisting of all $k$-dimensional compact convex subsets. In this paper we explore the topology of $K_k^n$ and $K_{k,b}^n$ and the relation of these hyperspaces with the Grassmann manifold $G_k(n)$. We prove that both $K_k^n$ and $K_{k,b}^n$ are Hilbert cube manifolds with a fiber bundle structure over $G_k(n)$. We also show that the fiber of $K_{k,b}^n$ with respect to this fiber bundle structure is homeomorphic with $\mathbb R^{\frac{k(k+1)+2n}{2}}\times Q$, where $Q$ stands for the Hilbert cube. |
| title | The hyperspace of k-dimensional closed convex sets |
| topic | General Topology Geometric Topology Primary: 52A20, 52A21, 54B20, 54C10, 54C55. Secondary: 54H15, 57S25 |
| url | https://arxiv.org/abs/2410.00839 |