Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access August 8, 2022

Branching Geodesics of the Gromov-Hausdorff Distance

  • Yoshito Ishiki EMAIL logo

Abstract

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

References

[1] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.10.1090/gsm/033Search in Google Scholar

[2] S. Chowdhury and F. Mémoli, Explicit geodesics in Gromov-Hausdorff space, Electron. Res. Announc. Math. Sci. 25 (2018), 48–59.Search in Google Scholar

[3] G. David and S. Semmes, Fractured fractals and broken dreams: Self similar geometry through metric and measure, Oxford Lecture Ser. Math. Appl., vol. 7, Oxford Univ. Press, 1997.Search in Google Scholar

[4] J. M. Fraser, Assouad dimension and fractal geometry, Tracts in Mathematics Series, vol. 222, Cambridge University Press, 2020.10.1017/9781108778459Search in Google Scholar

[5] J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001.10.1007/978-1-4613-0131-8Search in Google Scholar

[6] Y. Ishiki, Quasi-symmetric invariant properties of Cantor metric spaces, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 6, 2681–2721.Search in Google Scholar

[7] –––––, An interpolation of metrics and spaces of metrics, (2020), preprint, arXiv:2003.13277.Search in Google Scholar

[8] –––––, On dense subsets in spaces of metrics, (2021), arXiv:2104.12450, to appear in Colloq. Math.Search in Google Scholar

[9] A. Ivanov, S. Iliadis, and A. Tuzhilin, Realizations of Gromov–Hausdorff distance, preprint arXiv:1603.08850 (2016).Search in Google Scholar

[10] A. O. Ivanov, N. K. Nikolaeva, and A. A. Tuzhilin, The Gromov-Hausdorff metric on the space of compact metric spaces is strictly intrinsic, Mat. Zametki 100 (2016), no. 6, 947–950, translation in Math. Notes 100 (2016), no. 5-6, 883-885.Search in Google Scholar

[11] D. Jansen, Notes on pointed Gromov–Hausdorff convergence, (2017), arXiv:1703.09595.Search in Google Scholar

[12] D. P. Klibus, Convexity of a ball in the Gromov–Hausdorff space, Moscow Univ. Math. Bull. 73 (2018), no. 6, 249–253, Translation of Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2018, no. 6, 41–45.Search in Google Scholar

[13] F. Mémoli and Z. Wan, Characterization of Gromov-type geodesics, (2021), preprint, arXiv:2105.05369.Search in Google Scholar

[14] J. Rouyer, Generic properties of compact metric spaces, Topology Appl. 158 (2011), no. 16, 2140–2147.Search in Google Scholar

[15] K. Sakai, Topology of infinite-dimensional manifolds, Springer, 2020.10.1007/978-981-15-7575-4Search in Google Scholar

[16] Z. Wan, A novel construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric, Topology Appl. 300 (2021), 107759.10.1016/j.topol.2021.107759Search in Google Scholar

Received: 2021-08-30
Accepted: 2022-04-13
Published Online: 2022-08-08

© 2022 Yoshito Ishiki, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/agms-2022-0136/html
Scroll to top button