A topological insight into the polar involution of convex sets

Abstract 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, d₀ₖ) (K 0 n, d A W), where d AW denotes the Attouch–Wets metric. We prove that (K₀ⁿ, d₀ₖ) (K 0 n, d A W) is homeomorphic to the Hilbert cube Q = ₈ = 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.