Figure 1.
Branching geodesics as described in §1.1.2
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On the norm continuity of the hk-fourier transform
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 |
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 $n\geq 1$.
Mathematics Subject Classification:
Primary 53C23, Secondary 51F99.
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
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A. Hatcher,
Algebraic Topology, Cambridge University Press, Cambridge, 2002. |
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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,
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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). |
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