The hyperspace of k-dimensional closed convex sets

For every n >= 2, let K n k denote the hyperspace of all k-dimensional closed convex subsets of the Euclidean space Rn endowed with the Atouch-Wets topology. Let K nk,b be the subset of K n k consisting of all k-dimensional compact convex subsets. In this paper we explore the topology of K n k and K nk,b and the relation of these hyperspaces with the Grassmann manifold G k ( n ). We prove that both K n k and K nk,b are Hilbert cube manifolds with a fiber bundle structure over G k ( n ). We also show that the fiber of K nk,b with respect to this fiber bundle structure is homeomorphic with R k ( k +1)+2 n 2 x Q , where Q stands for the Hilbert cube. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).