| Abstract: |
| Tools from topology can bound or compute quantities arising in metric geometry. An example geometric quantity is the Hausdorff or Gromov-Hausdorff distance between two metric spaces (or datasets). Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. An example topological tool is the nerve lemma: a good cover of a space faithfully encodes the shape of that space. When X is a sufficiently dense subset of a closed Riemannian manifold M, we can use the nerve lemma to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved, and even obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) when M is the circle. |
|