Vegter, Gert
Gert
Vegter
Wintraecken, Mathijs
Mathijs
Wintraecken
The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds
2019
2019-07-12T08:34:57Z
2020-01-16T12:38:06Z
conference
https://research-explorer.app.ist.ac.at/record/6628
https://research-explorer.app.ist.ac.at/record/6628.json
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Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space.