A topological insight into the polar involution of convex sets

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 $\mathbb R^n$.