An introduction to the Gromov-Hausdorff distance

Title: An introduction to the Gromov-Hausdorff distance

Abstract: We give a brief introduction to the Hausdorff and Gromov-Hausdorff distances between metric spaces. The Hausdorff distance is defined on two subsets of a common metric space. The Gromov-Hausdorff distance is defined on any two arbitrary metric spaces. The Gromov-Hausdorff distance gives a metric on all isometry classes of compact metric spaces.

[sethjchandler]

I appreciate the effort to create a visual explanation of a complex concept. I think it would be better however, at least for those of us not versed in topology, To be a little clearer about what you mean by thickening. I also think that showing the numerical value of epsilon would make the initial example clearer. But, I realize this is a hard concept and that visualization is difficult. Thanks for the effort.

[bryanbischof]

Very clever use of the technology to add intuition. What are good algorithms for approximating Hausdorff dimension? GH seems pretty incomputable considering it’s over all isotopic embeddings.

[inothernews]

Thanks for the wonderful content. The clarity encourages viewers to further read more into the subject. Would love to see more!
I'm not so familiar with isometric embeddings, so I have some questions!
1. Will there always be a common Z into which X and Y may isometrically embed?
2. If there is such a Z, does it mean GH(X, Y) <= H(X, Y)? --- following the intuition that GH allows for X and Y to get as close to each other as possible while H does not. For example, even if X and Y are in the same metric space, GH allows them to be translated to get 'as near as possible'.
Thank you!

[ajigao]

Thanks for your vivid presentation.
I wonder if there is a mathematical relationship between Hausdorff measure and Hausdorff distance? I Intuitively thought that Hausdorff distance is the difference of two sets's Hausdorff measure, but I can‘t find the mathematical evidence. And where I can find more information about the derivation of Hausdorff distance?