Hyers–Ulam stability of bijective $$\varepsilon $$-isometries between Hausdorff metric spaces of compact convex subsets

Abstract

Let X (resp. Y) be a real Banach space such that the set of all $$w^*$$ -exposed points of the closed unit ball $$B(X^*)$$ (resp. $$B(Y^*)$$ ) is $$w^*$$ -dense in the unit sphere $$S(X^*)$$ (resp. $$S(Y^*)$$ ), (cc(X), H) (resp. (cc(Y), H)) be the metric space of all nonempty compact convex subsets of X (resp. Y) endowed with the Hausdorff distance H, and $$f:(cc(X),H)\rightarrow (cc(Y),H)$$ be a standard bijective $$\varepsilon $$ -isometry. Then there is a standard surjective isometry $$g:cc(X)\rightarrow cc(Y)$$ satisfying that $$(1)\, g|_{X}$$ (the restriction of g on $$\{\{u\}, u\in X\}$$ ) is a surjective linear isometry from $$\{\{u\},u\in X\}$$ onto $$\{\{v\},v\in Y\}$$ and $$g(A)=\cup _{a\in A}g|_{X}(\{a\})$$ for any $$A\in cc(X)$$ ; $$(2)\, H(f(A),g(A))\le 3\varepsilon $$ for any $$A\in cc(X)$$ .