A lower bound for splines on tetrahedral vertex stars
share
A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra – we call these closed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of C^r splines on closed vertex stars of degree at least 3r+2. We show that this formula is a lower bound on the dimension of C^r splines of degree at least (3r+2)/2. Our proof uses apolarity and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We also use an argument of Whiteley to show that the only splines of degree at most (3r+1)/2 on a generic closed vertex star are global polynomials.
READ FULL TEXT