A topological insight into the polar involution of convex sets

TL;DR

This paper explores the topological structure of the polar set mapping on convex sets containing the origin, revealing it is topologically equivalent to a Hilbert cube and characterizing related involutions.

Contribution

It establishes the homeomorphism between the space of convex sets with the Attouch-Wets metric and the Hilbert cube, and characterizes involutions with a unique fixed point as polar mappings composed with linear isomorphisms.

Findings

The space of convex sets is homeomorphic to the Hilbert cube.

The polar mapping is topologically conjugate to the standard involution on the Hilbert cube.

Unique fixed point involutions are characterized as polar mappings composed with positive definite linear transformations.

Abstract

Denote by $\mathcal{K}_0^n$ the family of all closed convex sets $A\subset\mathbb{R}^n$ containing the origin $0\in\mathbb R^n$. For $A\in\mathcal{K}_0^n,$ its polar set is denoted by $A^\circ.$ In this paper, we investigate the topological nature of the polar mapping $A\to A^\circ$ on $(\mathcal{K}_0^n, d_{AW})$, where $d_{AW}$ denotes the Attouch-Wets metric. We prove that $(\mathcal{K}_0^n, d_{AW})$ is homeomorphic to the Hilbert cube $Q=\prod_{i=1}^{\infty}[-1,1]$ and the polar mapping is topologically conjugate with the standard based-free involution $\sigma:Q\rightarrow Q,$ defined by $\sigma(x)=-x$ for all $x\in Q.$ We also prove that among the inclusion-reversing involutions on $\mathcal K^n_0$ (also called dualities), those and only those with a unique fixed point are topologically conjugate with the polar mapping, and they can be characterized as all the maps $f:\mathcal{K}_0^n\to \mathcal{K}_0^n$ of the form $f(A)=T(A^{\circ})$, with $T$ a positive definite linear isomorphism of $\mathbb R^n$.