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

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 epsilon-isometric embedding with epsilon >= 0, sigma(A)(x(*)) = sup(a is an element of A)(x(*), a), V A is an element of (cc(X) , H) and x(*) E B(X-*), and sigma f (A)(y(*)) = sup(y is an element of f(A))(y(*) , y) , V A E (cc(X), H) and y(*) is an element of B(Y-*). We show that, for instance, (1) if X is Gateaux smooth, then for any phi is an element of<((sigma(cc(X)) - sigma(cc(X))))over bar>(star) there exists psi is an element of ((span) over bar[sigma(f(cc(X))) - sigma(f(cc(X)))] with II psi II = II phi II = gamma such that | (psi, sigma f(A) - sigma f(B)) - (phi, sigma A - sigma B) |< 6 gamma epsilon, for all A, B E cc(X); (2) if X is Gateaux smooth and epsilon = 0, then there exists a unique surjective bounded linear operator F : (span) over bar[sigma(f(cc(X))) - sigma(f(cc(X)))] -> <(sigma(cc)(X) - sigma(cc(X)))over bar> with parallel to F parallel to = 1 such that F(sigma f(A) - sigma f({0})) = sigma A , V A is an element of cc(X); (3) if X is Gateaux smooth, then there exists a unique surjective linear isometric embedding g : <(sigma(cc)(X) - sigma(cc(X)))over bar> -> (span) over bar [sigma(f(cc(X))) - sigma(f(cc(X)))] such that parallel to g(sigma A) - (sigma(f(A)) - sigma f({0}))parallel to <= 6 epsilon, V A is an element of cc(X), if and only if for any mu is an element of (span) over bar sigma(f(cc(X))) - sigma(f(cc(X)))] with parallel to mu parallel to = 1, lim inf dist(t mu, sigma f(cc(X)) - sigma f(cc(X)))/|t|< 1/2; (4) if X and Y are Banach spaces of the same finite dimension, epsilon = 0, and f (A + B) = f (A) + f (B) for any pair A, B is an element of cc(X), then there is a surjective linear isometric embedding f : X -> Y such that f (A) = {(f) over bar (a) : a is an element of A}, for all A is an element of cc(X), where sigma(cc)(X)-sigma(cc)(X)equivalent to{sigma(A)-sigma(B):A,B is an element of cc(X)},sigma(f)(cc(X))-sigma f((cc(X)))equivalent to{sigma(f)(A)-sigma(f)(B):A,B is an element of cc(X)}, and dist (t mu,sigma(f) (cc(X))-sigma(f(cc(X)))) equivalent to inf {parallel to t mu - v parallel to : v is an element of sigma(sigma f(cc(X)))-sigma(f(cc(X)))}. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paragraph mainly discusses the properties and relationships on real Banach spaces, including embeddings, linear operators, isometric embeddings, etc.