On non-surjective ε-isometric embeddings between Hausdorff metric spaces of compact convex subsets

Abstract

Let X be a real Banach space, (cc(X),H) be the metric space of all non-empty compact convex subsets of X equipped with the Hausdorff metric H, f:(cc(X),H)→(cc(Y),H) be an ε-isometric embedding with ε≥0, σA(x⁎)=supa∈A⁡〈x⁎,a〉, ∀A∈(cc(X),H) and x⁎∈B(X⁎), and σf(A)(y⁎)=supy∈f(A)⁡〈y⁎,y〉, ∀A∈(cc(X),H) and y⁎∈B(Y⁎). We show that, for instance,(1)if X is Gâteaux smooth, then for any ϕ∈(σcc(X)−σcc(X)‾)⁎ there exists ψ∈(span‾[σf(cc(X))−σf(cc(X))])⁎ with ‖ψ‖=‖ϕ‖≡γ such that|〈ψ,σf(A)−σf(B)〉−〈ϕ,σA−σB〉|≤6γε,∀A,B∈cc(X);(2)if X is Gâteaux smooth and ε=0, then there exists a unique surjective bounded linear operator F:span‾[σf(cc(X))−σf(cc(X))]→σcc(X)−σcc(X)‾ with ‖F‖=1 such thatF(σf(A)−σf({0}))=σA,∀A∈cc(X);(3)if X is Gâteaux smooth, then there exists a unique surjective linear isometric embedding g:σcc(X)−σcc(X)‾→span‾[σf(cc(X))−σf(cc(X))] such that‖g(σA)−(σf(A)−σf({0}))‖≤6ε,∀A∈cc(X), if and only if for any μ∈span‾[σf(cc(X))−σf(cc(X))] with ‖μ‖=1,liminf|t|→∞dist(tμ,σf(cc(X))−σf(cc(X)))/|t|<1/2;(4)if X and Y are Banach spaces of the same finite dimension, ε=0, and f(A+B)=f(A)+f(B) for any pair A,B∈cc(X), then there is a surjective linear isometric embedding f‾:X→Y such thatf(A)={f‾(a):a∈A},∀A∈cc(X), where σcc(X)−σcc(X)≡{σA−σB:A,B∈cc(X)}, σf(cc(X))−σf(cc(X))≡{σf(A)−σf(B):A,B∈cc(X)}, and dist(tμ,σf(cc(X))−σf(cc(X)))≡inf⁡{‖tμ−v‖:v∈σf(cc(X))−σf(cc(X))}.