It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:58:04Z http://mathoverflow.net/feeds/question/23061 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? Paul Siegel 2010-04-30T02:18:31Z 2010-06-13T09:09:31Z <p>Recall that a space is $\delta$-hyperbolic if there is some number $\delta$ with the property that every point on an edge of a geodesic triangle lies within $\delta$ of another edge. For example a tree is $0$-hyperbolic. One of the basic facts about standard hyperbolic space is that it is $\delta$-hyperbolic for some $\delta$, and I am looking for the smallest delta which makes this true.</p> <p>Full disclosure: I stole this question from Dima Burago, who brought it up as an example of of a useless problem about which he is nevertheless a little curious. I haven't exactly burned the midnight oil, but I can't solve it.</p> http://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta/23063#23063 Answer by Matthew Stover for It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? Matthew Stover 2010-04-30T02:38:49Z 2010-04-30T03:05:44Z <p>Let $T$ be a triangle in $\mathbb{H}^2$. Its area is $\pi - \alpha - \beta - \gamma$, where $\alpha$, $\beta$, and $\gamma$ are the interior angles. You can find how slim this triangle is by considering an inscribed circle in $T$. The radius of this triangle, thus $\delta$, are bounded above by the area, so to find the $\delta$ that works for all triangles, you take the limit and consider an ideal triangle $T_\infty$. You can explicitly compute that the inscribed circle minimizing distance between the sides has length $4 \log \phi$, where $\phi$ is the golden ratio. (See <a href="http://en.wikipedia.org/wiki/Ideal_triangle" rel="nofollow">here</a> and <a href="http://392c.wordpress.com/2009/02/12/more-quasi-isometry-and-hyperbolic-metric-spaces/" rel="nofollow">here</a>.)</p> http://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta/23064#23064 Answer by S. Carnahan for It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? S. Carnahan 2010-04-30T03:17:54Z 2010-04-30T03:56:41Z <p>We can use the isometry group of $H^n$ to reduce to the case of an ideal triangle in the upper half plane, with vertices at -1, 1, and infinity. We want to find the distance between $i$ and the vertical geodesic with real part 1. To find the shortest geodesic, we reflect $i$ in the vertical line, and take half the distance between $i$ and $i+2$. The <a href="http://en.wikipedia.org/wiki/Poincare_metric" rel="nofollow">distance formula</a> yields $\tanh^{-1}\left(\frac{|(i+2)-i|}{|(i+2) + i|} \right) = \tanh^{-1}(1/\sqrt2)$. This is about 0.8813735.</p> http://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta/23068#23068 Answer by Sam Nead for It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? Sam Nead 2010-04-30T03:41:26Z 2010-04-30T03:41:26Z <p>Consider the ideal triangle with vertices at infinity, zero, and one. Let $C$ be the semicircle perpendicular to the vertical line $[0, \infty]$ and meeting $1/2 + i/2$ (ie the midpoint of the semicircle $[0,1]$). So $C$ meets $[0, \infty]$ at the point $i \cdot \sqrt{2}/2$. Scale down by a factor of $\sqrt{2}/2$ to get the point $1/\sqrt{2} + i/\sqrt{2}$. Use a Mobius transformation to rotate this by $\pi/2$ about $i$ to get $i(1+\sqrt{2})$. Now integrate $dy/y$ to get $\log(1 + \sqrt{2})$. This is approximately 0.88137358702.</p> http://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta/28007#28007 Answer by Junyan Xu for It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? Junyan Xu 2010-06-13T09:09:31Z 2010-06-13T09:09:31Z <p>See <a href="http://www.math.umn.edu/~am/book/outercircles.pdf" rel="nofollow">http://www.math.umn.edu/~am/book/outercircles.pdf</a>, p.14-15 for detailed treatment.</p>