🤖 AI Summary
This work addresses the ambiguity in simplex diameter shrinkage during iterative refinement of spherical Delaunay complexes, which arises from their non-nested structure. To resolve this issue, the authors introduce several Steiner point insertion strategies that recast the shrinkage analysis as a covering estimation problem for Euclidean simplices. By proposing a sharp, dimension-dependent variant of the approximate Carathéodory theorem and integrating a key sampling technique, they establish, for the first time, an explicit bound on diameter shrinkage in Delaunay refinement. Both theoretical analysis and numerical experiments demonstrate that the proposed approach achieves stronger and quantifiable simplex shrinkage compared to conventional subdivision strategies.
📝 Abstract
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.