On the Optimal Triangulation of Convex Hypersurfaces, Whose Vertices Lie in Ambient Space

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 (Formula presented.) an optimal triangulation of Σ with m vertices. Here we use optimal in the sense of minimizing dH(Sm,Σ), where dH 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 dH(Sm,Σ) in m is twice that of (Formula presented.). This statement is proven.