The hyperspace of k-dimensional closed convex sets

Adriana Escobedo-Bustamante; Natalia Jonard-Pérez

paper

The hyperspace of k-dimensional closed convex sets

arXiv:2410.00839

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.