A topological insight into the polar involution of convex sets

Abstract

AbstractDenote by $${\cal K}_0^n$$ 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 ∈ 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 $$({\cal K}_0^n,{d_{AW}})$$ ( K 0 n , d A W ) , where dAW denotes the Attouch–Wets metric. We prove that $$({\cal K}_0^n,{d_{AW}})$$ ( K 0 n , d A W ) is homeomorphic to the Hilbert cube $$Q = \prod\nolimits_{i = 1}^\infty {[ - 1,1]} $$ 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 $${\cal K}_0^n$$ 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 : K 0 n → K 0 n of the form f(A) = T(A°), with T a positive-definite linear isomorphism of ℝn.