Note on $(m,q)$-isometries on an hyperspace of a normed space

Abstract

Given a normed space $X$ we consider the hyperspace $k(X)$ of all non-empty compact convex subsets of $X$ endowed with the Hausdorff distance. We prove that if $T: X \longrightarrow X$ is an $(m,q)$-isometry, then it is possible that the map $k(T) : k(X) \longrightarrow k(X)$, $k(T) C := TC$, is not an $(m,q)$-isometry. Moreover, if $\widehat{k(X)}$ is the R{\aa}dstr\"{o}m space associated to the hyperspace $k(X)$, then $\mathcal{T}: k(X) \longrightarrow k(X)$ is an $(m,q)$-isometry if and only if $\widehat{\mathcal{T}}: \widehat{k(X)} \longrightarrow \widehat{k(X)}$ is an $(m,q)$-isometry.