A topological insight into the polar involution of convex sets

Denote by \({\cal K}_0^n\) $${\mathcal{K}}_0^n$$ the family of all closed convex sets A ⊂ ℝⁿ containing the origin 0 ∈ ℝⁿ. For \(A \in {\cal K}_0^n\) $$A\in {\mathcal{K}}_0^n$$ , its polar set is denoted by A°. In this paper, we investigate the topological nature of the polar mapping A → A° on \(({\cal K}_0^n,{d_{AW}})\) $$({\mathcal{K}}_0^n,d_{AW})$$ , where d_AW denotes the Attouch–Wets metric. We prove that \(({\cal K}_0^n,{d_{AW}})\) $$({\mathcal{K}}_0^n,d_{AW})$$ is homeomorphic to the Hilbert cube \(Q = \prod\nolimits_{i = 1}^\infty {[ - 1,1]} \) $$Q={\displaystyle {\prod }_{i=1}^{\infty }[-1,1]}$$ and the polar mapping is topologically conjugate with the standard based-free involution σ: Q → Q, defined by σ(x) = −x for all x ∈ Q. We also prove that among the inclusion-reversing involutions on \({\cal K}_0^n\) $${\mathcal{K}}_0^n$$ (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 \(f:{\cal K}_0^n \to {\cal K}_0^n\) $$f:{\mathcal{K}}_0^n\to {\mathcal{K}}_0^n$$ of the form f(A) = T(A°), with T a positive-definite linear isomorphism of ℝⁿ.