Some \((n-1)\)-dimensional rigid sets in \(\mathbb{R}^ n\)
The author considers subsets of Euclidean \(n\)-space which are geometrically acceptable as metric spaces in the sense of \textit{K. Borsuk} [Bull. Pol. Acad. Sci., Math. 29, 83-90 (1981; Zbl 0459.54020)]. Such a subset is called rigid if every intrinsic isometry is the restriction of an isometry of the ambient space. Here it is shown that rigidity of a subset is preserved if a countable number of points are removed from its interior. This leads to a construction of \((n-1)\)-dimensional rigid subsets of Euclidean \(n\)-space. The examples given in this paper are non- compact.
scientific article; zbMATH DE number 569283
scientific article