2018, 25: 48-59. doi: 10.3934/era.2018.25.006

Explicit geodesics in Gromov-Hausdorff space

Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210. Phone: (614) 292-4975, Fax: (614) 292-1479

Received  March 31, 2017 Revised  March 13, 2018 Published  June 2018

Fund Project: This work was supported by NSF grants CCF-1526513 and IIS-1422400.

We provide an alternative, constructive proof that the collection ${\mathcal{M}}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on ${\mathcal{M}}$. We also provide several interesting examples of geodesics on ${\mathcal{M}}$, including a geodesic between ${\mathbb{S}}^0$ and ${\mathbb{S}}^n$ for any $n\geq 1$.

Citation: Samir Chowdhury, Facundo Mémoli. Explicit geodesics in Gromov-Hausdorff space. Electronic Research Announcements, 2018, 25: 48-59. doi: 10.3934/era.2018.25.006
References:
[1]

B. Bollobás, The Art of Mathematics: Coffee time in Memphis, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511816574.  Google Scholar

[2]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.  Google Scholar

[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, AMS Graduate Studies in Math., 33, American Mathematical Society, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999.  Google Scholar

[5]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[6]

A. Ivanov, N. Nikolaeva and A. Tuzhilin, The Gromov-Hausdorff metric on the space of compact metric spaces is strictly intrinsic, (Russian) Mat. Zametki, 100 (2016), 947-950; translation in Math. Notes, 100 (2016), 883-885.  Google Scholar

[7]

V. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, 40, American Mathematical Soc., Providence, RI, 2006.  Google Scholar

[8]

P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006.  Google Scholar

[9]

K.-T. Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, preprint, arXiv: 1208.0434, (2012). Google Scholar

show all references

References:
[1]

B. Bollobás, The Art of Mathematics: Coffee time in Memphis, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511816574.  Google Scholar

[2]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.  Google Scholar

[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, AMS Graduate Studies in Math., 33, American Mathematical Society, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999.  Google Scholar

[5]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[6]

A. Ivanov, N. Nikolaeva and A. Tuzhilin, The Gromov-Hausdorff metric on the space of compact metric spaces is strictly intrinsic, (Russian) Mat. Zametki, 100 (2016), 947-950; translation in Math. Notes, 100 (2016), 883-885.  Google Scholar

[7]

V. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, 40, American Mathematical Soc., Providence, RI, 2006.  Google Scholar

[8]

P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006.  Google Scholar

[9]

K.-T. Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, preprint, arXiv: 1208.0434, (2012). Google Scholar

Figure 1.  Branching geodesics as described in §1.1.2
[1]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[2]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[3]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

2019 Impact Factor: 0.5

Article outline

Figures and Tables

[Back to Top]