Abstract
Let \({\Sigma}\) be a strictly convex (hyper-)surface, S m an optimal triangulation (piecewise linear in ambient space) of \({\Sigma}\) whose m vertices lie on \({\Sigma}\) and \({\tilde{S}_m}\) an optimal triangulation of \({\Sigma}\) with m vertices. Here we use optimal in the sense of minimizing \({d_H(S_m, \Sigma)}\), where \({d_H}\) denotes the Hausdorff distance. In ‘Lagerungen in der Ebene, auf der Kugel und im Raum’ Fejes Tóth conjectured that the leading term in the asymptotic development of \({d_H(S_m, \Sigma)}\) in m is twice that of \({d_H(\tilde{S}_m, \Sigma)}\). This statement is proven.
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Abbreviations
- A d :
-
Fejes Tóth’s approximation parameter
- CH:
-
Convex hull of a subset of \({\mathbb{R}^d}\)
- C :
-
Convex body
- d H :
-
Hausdorff distance
- \({\kappa_d}\) :
-
Volume of the d-dimensional ball, that is \({\pi ^{d/2} / \Gamma (1+d/2)}\).
- K :
-
Gaussian curvature
- L :
-
Line segment contained in a hypersurface
- m :
-
Number of vertices
- m t :
-
Number of triangles (only 2 dimensional case)
- OL p :
-
Hyperplane orthogonal to the line segment L going through \({p \in L}\)
- P m :
-
Polygon/polytope with m vertices
- \({P_m^{{\rm in}}}\) :
-
Inscribed polygon/polytope with m vertices
- S m :
-
Simplicial complex with m vertices
- \({S_m^{{\rm on}}}\) :
-
Simplicial complex whose m vertices lie on the hypersurface (often at a stage where the complex is not yet prover to be convex)
- \({\Sigma}\) :
-
Hypersurface
- \({\theta_d}\) :
-
The optimal covering density of Euclidean space by unit balls
- \({U(X,\epsilon) }\) :
-
\({\epsilon}\) neighbourhood of X
- \({v_i, \tilde{v}_i}\) :
-
Vertices
- V i :
-
Set of vertices
References
Federer H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Fejes Tóth L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin, Göttingen, Heidelberg (1953)
Gruber P.M.: Assymptotic estimates for best and stepwise approximation of convex bodies I. Forum Math. 5, 281–297 (1993)
Gruber P.M.: Assymptotic estimates for best and stepwise approximation of convex bodies II. Forum Math. 5, 521–538 (1993)
Munkres J.R.: Topology. Prentice-Hall, Upper Saddle River (2000)
Rogers C.A.: Packing and Covering. Cambridge University Press, Cambridge (1964)
Schneider R.: Zur optimalen approximation konvexer hyperflächen durch polyeder. Math. Ann. 256, 289–301 (1981)
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Wintraecken, M.H.M.J., Vegter, G. On the Optimal Triangulation of Convex Hypersurfaces, Whose Vertices Lie in Ambient Space. Math.Comput.Sci. 9, 345–353 (2015). https://doi.org/10.1007/s11786-014-0216-7
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DOI: https://doi.org/10.1007/s11786-014-0216-7