\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Explicit geodesics in Gromov-Hausdorff space

This work was supported by NSF grants CCF-1526513 and IIS-1422400

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Branching geodesics as described in §1.1.2

  • [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).
  • 加载中

Figures(1)

SHARE

Article Metrics

HTML views(2726) PDF downloads(270) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return