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Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ where $$ \{x,y\}\in \mathcal{E}_k \text{ iff } d(x,y)\le k\delta $$ and where $$ W(\{x,y\})=k\delta \text{ iff } (k-1)\delta < d(x,y)\le k\delta . $$

Let $d_{k,\delta}$ denote the shortest path distance on $G_k$ given by $$ d_{k,\delta}(x,y) := \min \sum_{((u_i,u_{i+1}))_{i=1}^L;\,x=u_1;\,u^{L+1}=y}\, W(u_i,u_{i+1}) $$ where we minimize over all paths from $x$ to $y$.

Does $(G_{k,\delta},d_{k,\delta})$ coverage to $(X,d)$ in the Gromov-Hausdorff (or some other meaningful) sense?

Intuitively I'm asking: How many nearby points do I need to connect, to ensure that I can "consistently" approximate a metric space by a sequence of graph "discretizations"?

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  • $\begingroup$ Don't the $\delta$-packings themselves already converge to $X$ in the Hausdorff metric on $X$ (and therefore the Gromov-Hausdorff metric)? $\endgroup$
    – James E Hanson
    Commented Sep 11 at 0:35
  • $\begingroup$ But the distance is being measured differently, its not exactly equal to the usual distance. Also, Im wondering if there is a stronger sense in which this convergence happens... $\endgroup$
    – Math_Newbie
    Commented Sep 11 at 0:36
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    $\begingroup$ Oh I see, I missed that part. $\endgroup$
    – James E Hanson
    Commented Sep 11 at 0:37
  • $\begingroup$ What are you doing with $k$ in terms of convergence? Are you keeping it fixed, or are you also varying it with $\delta$? $\endgroup$
    – James E Hanson
    Commented Sep 11 at 0:40
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    $\begingroup$ This is not a good example @MoisheKohan since the weight of each edge is exactly equal to the distance between those points. $\endgroup$
    – Math_Newbie
    Commented Sep 11 at 13:36

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