Gromov-Hausdorff convergence for locally finite spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:08:54Z http://mathoverflow.net/feeds/question/82784 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82784/gromov-hausdorff-convergence-for-locally-finite-spaces Gromov-Hausdorff convergence for locally finite spaces Valerio Capraro 2011-12-06T11:18:13Z 2011-12-06T13:50:21Z <p><strong>Update:</strong> I've edited the question, since maybe it was a bit confusing and it's better to start with a more basic question.</p> <p>I'm looking for properties of Gromov-Hausdorff convergence in the particular case when all metrics $d_n$ and $d$ are defined on a fixed finite or countable set $X$ and they give rise to locally finite spaces.</p> <p>Fix a point $x\in X$ and denote by $R_x^n$ (resp. $R_x$) the radius of the smallest closed ball, in the metric $d_n$ (resp. $d$), about $x$ which contains at least two points. I suppose that every $(X,d_n)$ is $\delta_n$-connected; i.e. $\delta_n$ is a non-negative real number such that</p> <ol> <li><p>$|R_x^n-R_y^n|\leq\delta_n$, for all $x,y\in X$</p></li> <li><p>If $\delta'\geq0$ verifies $|R_x^n-R_y^n|\leq\delta'$, for all $x,y\in X$, then $\delta'\geq\delta_n$</p></li> <li><p>For all $x,y\in X$, there is a $\delta_n$-connection; i.e. a finite sequence of points $x_0,x_1,\ldots,x_{n-1},x_n$ such that $x_0=x$, $x_n=y$ and $d(x_i,x_{i-1})\leq\min(R_{x_i}^n+\delta_n,R_{x_{i-1}}^n+\delta_n)$, for all $i=1,\ldots,n$.</p></li> </ol> <p>Example: connected graph are $0$-connected spaces. If you like, I am just studying deformations of graphs.</p> <blockquote> <p><strong>Question:</strong> Suppose that $(X,d_n)$ is a sequence of $\delta_n$-connected spaces which converges in Gromov-Hausdorff sense to some $(X,d)$. Is it true that $\delta_n\rightarrow\delta$ and $(X,d)$ is $\delta$-connected?</p> </blockquote> <p>Thank you in advance,</p> <p>Valerio</p>