The author studies the metric structure of \((F(X),d_ H)\) where \((F(X),d_ H)\) denotes the space of compact non-empty subsets of a metric space (X,d) equipped with the Hausdorff metric. In a series of papers \textit{P. M. Gruber} and \textit{R. Tichy} [e.g. Monatsh. Math. 93, 116-126 (1982; Zbl 0449.52001)] have shown that for several important metric spaces (X,d) the only isometries from \((F(X),d_ H)\) into itself are those induced by the d-isometries of the basic space (X,d). The author tries to explain results of this kind from a more general point of view. His methods apply to a wide range of metric spaces and isometries and even Lipschitz maps. Moreover, as the author observes himself, it seems likely that further interesting generalizations of his results can be obtained. His main point is that the metric structure of an \(\epsilon\)-neighborhood of a point A in F(X) depends on the cardinality of A as a subset of X.