Let Σ be a strictly convex (hyper-)surface, Sm an optimal triangulation (piecewise linear in ambient space) of Σ whose m vertices lie on Σ and Sm an optimal triangulation of Σ with m vertices. Here we use optimal in the sense of minimizing d_H (Sm, Σ), where d_H denotes the Hausdorff distance. In ‘Lagerungen in der Ebene, auf der Kugel und im Raum' Fejes Toth conjectured that the leading term in the asymptotic development of d_H (Sm, Σ) in m is twice that of d_H(Sm, Σ). This statement is proven.
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机译:设Σ为严格凸(超)曲面,Sm为Σ的最佳三角剖分(在环境空间中呈分段线性),其中m个顶点位于Σ上,而Sm为Σ的最佳三角剖分。在这里,我们使用d_H(Sm,Σ)最小的最佳值,其中d_H表示Hausdorff距离。在“ Ebene的拉格伦根”中,auf der Kugel und im Raum的Fejes Toth猜想m中的d_H(Sm,Σ)渐近发展的领先项是d_H(Sm,Σ)的两倍。
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