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

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.