The hyperspace of k-dimensional closed convex sets

TL;DR

This paper investigates the topological structure of hyperspaces of k-dimensional convex sets in Euclidean space, revealing they are Hilbert cube manifolds with a fiber bundle structure over Grassmannians.

Contribution

It establishes that these hyperspaces are Hilbert cube manifolds with a fiber bundle structure over Grassmann manifolds, detailing the fiber's homeomorphism type.

Findings

Both hyperspaces are Hilbert cube manifolds.

They have a fiber bundle structure over Grassmann manifolds.

The fiber is homeomorphic to ^{(k(k+1)+2n)/2} imes Q.

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 $\mathbb R^{\frac{k(k+1)+2n}{2}}\times Q$, where $Q$ stands for the Hilbert cube.