Sharp approximate Carathéodory theorem and application to iterated Delaunay refinement
Provides theoretical guarantees for mesh contraction in Delaunay refinement, benefiting computational geometry and mesh generation communities.
The paper analyzes simplex diameter contraction under iterated Delaunay refinement, deriving explicit bounds for several Steiner point families. Theoretical and numerical results show Delaunay refinements achieve stronger contraction than subdivision counterparts.
We analyze the decrease of simplex diameters under iterated refinement of spherical Delaunay complexes. Unlike in ordinary subdivision, the refined Delaunay complex need not be a subdivision of the previous one, so mesh contraction is not automatic. We derive explicit contraction bounds for several families of Steiner points, including Delaunay analogues of barycentric and edgewise subdivision. The proof reduces the problem to sharp covering estimates for Euclidean simplices. These estimates are obtained through a strengthening of Maurey's empirical method via pivotal sampling and a dimension-dependent version of the approximate Carathéodory theorem. Theoretical results and numerical experiments show that Delaunay refinements achieve stronger contraction than their subdivision counterparts.