The hyperspace of k-dimensional closed convex sets

2024/10/01 by Escobedo-Bustamante, Adriana, Jonard-Pérez, Natalia
#52A21 #54B20 #54C10 #54C55. Secondary: 54H15 #57S25 #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Primary: 52A20

paper · doi:10.48550/arxiv.2410.00839

Abstract

For every n ≥ 2, let K_kⁿ denote the hyperspace of all k-dimensional closed convex subsets of the Euclidean space Rⁿ endowed with the Atouch-Wets topology. Let K_k,bⁿ be the subset of K_kⁿ consisting of all k-dimensional compact convex subsets. In this paper we explore the topology of K_kⁿ and K_k,bⁿ and the relation of these hyperspaces with the Grassmann manifold G_k(n). We prove that both K_kⁿ and K_k,bⁿ are Hilbert cube manifolds with a fiber bundle structure over G_k(n). We also show that the fiber of K_k,bⁿ with respect to this fiber bundle structure is homeomorphic with \mathbb R^{(k(k+1)+2n)/(2)}× Q, where Q stands for the Hilbert cube.

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