A Bound on the Number of Lines Through a Point in the Wang–Shi Split

This contribution answers the question that Tom Lyche addressed to the public of the 2022-INDAM meeting in Cortona “Approximation Theory and Numerical Analysis meet Algebra, Geometry, Topology” while presenting his work with Carla Manni and Hendrik Speleers. The question is if the number of lines through any point in the Wang–Shi split of degree d is always less than or equal to \(d+1\) . The question is purely geometric, but it has application to the construction of piecewise polynomial spaces with maximal order of continuity on general triangulations by splitting each triangle in sub-polygons.