Abstract
We find the necessary and sufficient conditions under which an unbounded metric space X has, at infinity, a unique pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) for every scaling sequence \( \tilde{r} \). In particular, it is proved that \( {\Omega}_{\infty, \tilde{r}}^X \) is unique and isometric to the closure of X for every logarithmic spiral X and every \( \tilde{r} \). It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the “asymptotic asymmetry” of these subsets.
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This paper dedicated to the memory of Professor Bogdan Bojarski
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 1, pp. 57–87 January–March, 2019.
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Dovgoshey, O., Bilet, V. Uniqueness of spaces pretangent to metric spaces at infinity. J Math Sci 242, 796–819 (2019). https://doi.org/10.1007/s10958-019-04517-1
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DOI: https://doi.org/10.1007/s10958-019-04517-1