Convex polyhedra in \(\mathbb{R}^3\) spanning \(\Omega(n^{4/3})\) congruent triangles
The authors construct, for any fixed triangle \(T\) and number \(n\) of vertices, an \(n\)-vertex convex polyhedron such that its vertices determine \(\Omega(n^{4/3})\) triangles congruent to \(T\). (In fact, the construction given is actually for a quadrilateral \(Q\) with perpendicular diagonals; the result for triangles follows trivially.) The construction begins with a family of planar configurations, due to Erdős, of \(n\) points and \(n\) lines that have \(cn^{4/3}\) incidences. These are used (following \textit{P. Erdős, D. Hickerson}, and \textit{J. Pach} [Am. Math. Mon. 96, 569-575 (1989; Zbl 0737.05006), here used as [1]]) to construct four small clusters of points on the unit sphere whose elements determine \(cn^{4/3}\) squares of diagonal 2 (imagine the square pivoting on vertices in the ``north'' and ``south'' clusters, with its other vertices tracing lines of the ``eastern'' and ``western'' clusters).f Finally, this is modified, by scaling each cluster appropriately around the origin, to yield a set that works for the quadrilateral \(Q\). This last adaptation, which is the novel element in this paper, is slightly trickier than it appears. The construction in [1] must be modified in order that the angles subtended by each cluster may be arbitrarily small, or else the four clusters taken together may contain points not on their convex hull.
scientific article; zbMATH DE number 1760689
scientific article