Partitions of planar sets into small triangles
For a set P of 3n points in a convex planar body \(\Sigma\), a(n,P,\(\Sigma)\) is the smallest sum of areas of n triangles formed by the points P, such that the vertices of the triangles exhaust the 3n points of P and the triangles are pairwise disjoint. If a(n,\(\Sigma)\) denotes the supremum of a(n,P,\(\Sigma)\) over all \(P\subset \Sigma\) of 3n points, and a(n) denotes the supremum of a(n,\(\Sigma)\)/A(\(\Sigma)\) over all convex planar sets \(\Sigma\) with positive area A(\(\Sigma)\), then for \(n\geq 2\) the following is established \[ n^{-1/2}\ll a(n)\leq 0.8+O(n^{-1})\leq 0.9. \] Open problem: What is the limit of a(n) for \(n\to \infty ?\)
scientific article; zbMATH DE number 3949171