A topological insight into the polar involution of convex sets

Abstract

Denote by \({\cal K}_0^n\) the family of all closed convex sets A ⊂ ℝⁿ containing the origin 0 ∈ ℝⁿ. For \(A \in {\cal K}_0^n\), its polar set is denoted by A°. In this paper, we investigate the topological nature of the polar mapping A → A° on \(({\cal K}_0^n,{d_{AW}})\), where d_AW denotes the Attouch–Wets metric. We prove that \(({\cal K}_0^n,{d_{AW}})\) is homeomorphic to the Hilbert cube \(Q = \prod\nolimits_{i = 1}^\infty {[ - 1,1]} \) 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 \({\cal K}_0^n\) (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 \(f:{\cal K}_0^n \to {\cal K}_0^n\) of the form f(A) = T(A°), with T a positive-definite linear isomorphism of ℝⁿ.