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arXiv · 2606.25854

Sharp approximate Carath\'eodory theorem and application to iterated Delaunay refinement

Raphaël Tinarrage
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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 Carath\'eodory theorem. Theoretical results and numerical experiments show that Delaunay refinements achieve stronger contraction than their subdivision counterparts.

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Raphaël Tinarrage. 2026-06-24. Sharp approximate Carath\'eodory theorem and application to iterated Delaunay refinement. https://arxiv.org/abs/2606.25854

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