This is in regards to the paper "When Lines go bad in hyperspace" by Christopher Bay, Amber Lembcke, and Steven Schlicker which appears in Demonstratio Mathematica, No. 3, Volume 38 (2005), p. 689-701. It has recently been brought to our attention that Theorem 1 from this paper is not correct. Please note that the main conclusions of the paper do not depend at all on this theorem. However, as the authors we feel it is our responsibility to bring this erroneous theorem to your attention. As stated in the paper, Theorem 1 intends to demonstrate that there can be infinitely many elements at a given location between two sets A and B.
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Let H(Rn) denote the hyperspace of all non-empty compact subsets of Rn. The Hausdorff metric h provides a way to measure distances between two elements of H(Rn) and generates the complete metric space (H(Rn),h)). In this paper, we examine geometric properties of lines in H(Rn), as determined by the Hausdorff metric, and compare and contrast the properties of these lines with Euclidean lines in Rn. Several surprising properties of these objects will be highlighted.
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