Hyers-Ulam stability of \(\varepsilon \)-isometries between the positive cones of \(c_0\) (Q2071837)

Let $X$ be a Banach space over the reals. A wedge $W$ in $X$ is a convex set which is closed under multiplication by non-negative scalars. The paper under review investigates the following problem: Let $X, Y$ be two Banach spaces, $W_1\subset X$, $W_2\subset Y$ be two wedges, $f\colon W_1\rightarrow W_2$ be a standard surjective $\varepsilon$-isometry. Do there exist a surjective (additive) isometry $V\colon W_1\rightarrow W_2$ and $\gamma>0$ such that $$ \|f(x)-V(x)\|\leq \gamma\varepsilon \text{ for all }x\in W_1? $$ The main result of the author proves that the above-mentioned problem has an affirmative answer when $X=Y=c_0$ and $W_1=W_2=c^+_0$, where $c^+_0$ is the positive cone of $c_0$. Moreover, the assumption on the surjectivity of $f$ can be relaxed by requiring $f$ to be $\delta$-surjective for some $\delta\geq 0$.