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

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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.

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

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

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

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.

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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).

Figure 1.  Branching geodesics as described in §1.1.2
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