A topological insight into the polar involution of convex sets

Denote by K₀ⁿ K 0 n the family of all closed convex sets A ⊂ ℝ n containing the origin 0 ∈ ℝ n . For A ∈ K₀ⁿ A ∈ 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 ( K₀ⁿ,dAW) ( K 0 n , d A W ) , where d AW denotes the Attouch–Wets metric. We prove that ( K₀ⁿ,dAW) ( K 0 n , d A W ) is homeomorphic to the Hilbert cube Q = ∏i = 1^∞ [ - 1,1] Q = ∏ i = 1 ∞ [ − 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 K₀ⁿ 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: K₀ⁿ → K₀ⁿ f : K 0 n → K 0 n of the form f ( A ) = T ( A °), with T a positive-definite linear isomorphism of ℝ n .