AbstractLet$(X,+)$be an Abelian group and$E$be a Banach space. Suppose that$f:X\rightarrow E$is a surjective map satisfying the inequality$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$for some${\it\varepsilon}>0$,$p>1$and for all$x,y\in X$. We prove that$f$is an additive map. However, this result does not hold for$0<p\leq 1$. As an application, we show that if$f$is a surjective map from a Banach space$E$onto a Banach space$F$so that for some${\it\epsilon}>0$and$p>1$$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$whenever$\Vert x-y\Vert =\Vert u-v\Vert$, then$f$preserves equality of distance. Moreover, if$\dim E\geq 2$, there exists a constant$K\neq 0$such that$Kf$is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,Studia Math.45(1973) 43–48].
Remark on our paper ``On bases in Banach spaces'' (Studia Math. 170 (2005), 147–171)
