Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets

RESUMO

Let $X$ and $Y$ be real Banach spaces, and let $\cC(X)$ and $\cC(Y)$ denote the families of all nonempty bounded closed convex subsets of $X$ and $Y$, respectively, equipped with the Hausdorff metric. We prove that every surjective isometry $F:\cC(X)\to\cC(Y)$ is induced by a surjective affine isometry of the underlying spaces. More precisely, $F$ maps singleton sets onto singleton sets, and the map $T:X\to Y$ defined by $F(\{x\})=\{T(x)\}$ is a surjective affine isometry satisfying \[ F(A)=T[A]:=\{T(x):\;x\in A\},\qquad(A\in\cC(X)). \] In particular, our theorem extends the result of Gruber and Lettl for finite-dimensional Euclidean spaces to arbitrary real Banach spaces, without imposing any additional assumptions on the underlying spaces.