Source for: Geodesics in CAT(0) spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:32:08Z http://mathoverflow.net/feeds/question/102197 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102197/source-for-geodesics-in-cat0-spaces Source for: Geodesics in CAT(0) spaces Joseph O'Rourke 2012-07-14T01:19:49Z 2012-07-14T09:22:34Z <p>I am seeking a good introductory reference that could lead to an understanding of the properties of geodesics in complete <a href="http://en.wikipedia.org/wiki/CAT%28k%29_space" rel="nofollow">CAT(0) metric spaces</a>. I am especially interested in learning the differences between geodesics in these spaces and those in an \$n\$-dimensional Euclidean space with its usual Euclidean metric, which is of course CAT(0). I am ultimately interested in simplicial, cubical, and polyhedral complexes, but I am willing to start anywhere. Thanks for educating me! <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://ars.els-cdn.com/content/image/1-s2.0-S0925772112000752-gr001.jpg" alt="alt text"> <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <sub>[Image from "Shortest path problem in rectangular complexes of global nonpositive curvature" (<a href="http://www.sciencedirect.com/science/article/pii/S0925772112000752" rel="nofollow">Elsevier link</a>)]</sub></p> http://mathoverflow.net/questions/102197/source-for-geodesics-in-cat0-spaces/102201#102201 Answer by Patricia Hersh for Source for: Geodesics in CAT(0) spaces Patricia Hersh 2012-07-14T01:55:05Z 2012-07-14T01:55:05Z <p>Your pictures remind me of the space of phylogenetric trees (which is cubical and exhibits similar folding), so if you are interested in reading about very concrete such examples, you might take a look at: </p> <ol> <li>L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, Advances in Applied Math, 27 (2001), no. 4, 733-467. </li> <li>M. Owen and S. Provan, A fast algorithm for computing geodesic distances in tree space, IEEE/ACM Trans. Computational Biology and Bioinformatics, 8: 2-13, 2011. </li> <li>F. Ardila, M. Owen and S. Sullivant, Geodesics in CAT(0) Cubical Complexes, Advances in Applied Math. 48 (2012), 142-163.</li> </ol> <p>(1) proves uniqueness of geodesics in tree space, (2) gives a polynomial time algorithm to find geodesic distances, and (3) works in more generality.</p> http://mathoverflow.net/questions/102197/source-for-geodesics-in-cat0-spaces/102212#102212 Answer by Igor Rivin for Source for: Geodesics in CAT(0) spaces Igor Rivin 2012-07-14T09:22:34Z 2012-07-14T09:22:34Z <p>The canonical reference is Bridson and Haefliger, Metric Spaces of non-positive curvature</p>