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The hyperspace of k-dimensional closed convex sets

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-01 DOI:10.1016/j.topol.2024.109154

Adriana Escobedo-Bustamante , Natalia Jonard-Pérez

引用次数: 0

Abstract

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

R^{\frac{k (k + 1) + 2 n}{2}} \times Q

, where Q stands for the Hilbert cube.

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k维闭凸集的超空间

当n≥2时，设Kkn表示欧几里德空间Rn中具有Atouch-Wets拓扑的所有k维闭凸子集的超空间。设Kk bn是Kkn的子集由所有k维紧凸子集组成。本文研究了Kkn和Kk，bn的拓扑结构，以及这些超空间与格拉斯曼流形Gk(n)的关系。证明了Kkn和Kk，bn都是Gk(n)上具有纤维束结构的希尔伯特立方流形。我们还证明了Kk bn的纤维束结构与Rk(k+1)+2n2×Q是同胚的，其中Q代表希尔伯特立方体。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.