Isometries between a metric space and its hyperspace, function space, and space of measures. (Q1426504)

The authors show that there does not exist a bounded metric space \(X\) with more than one element which is isometric to the metric space \(CL(X)\) of nonempty and closed subsets of \(X\) endowed with the Hausdorff metric, and there does not exist a bounded metric space \(X\) which is isometric to the metric space \(N(X)\) of nonexpansive real-valued functions on \(X\) endowed with the supremum metric. They leave open the question whether there exists a bounded metric space \(X\) with more than one element which is isometric to the metric space \(M(X)\) of Borel probability measures on \(X\) endowed with the Hutchinson metric. Their main theorem states that for every metric space \(X\) with more than one element, there is a \(\delta>0\) such that there is no expansive function from the set \(D_\delta(X)\) of nonempty and \(\delta\)-discrete subsets of \(X\) endowed with the Hausdorff metric to \(X.\) Problems about fixed points in theoretical computer science motivate the presented investigations.