  A tetrahedral complex all of whose tetrahedra meet at a common vertex is
called a \textit{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 \textit{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 \textit{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.
