Supersmoothness of the Alfeld Split

We consider spline functions over an Alfeld split in \({\mathbb R}^n\) , with smoothness r on the faces. In a recent paper, we derived the maximal order of supersmoothness of such a spline at the interior vertex. The proof was based on a characterization of supersmoothness in terms of the degeneracy of polynomial spline spaces on the split, and on the dimension formula for such spaces due to Foucart, Sorokina, and Schenck. In this note, we present an alternative, inductive proof of supersmoothness using only properties of directional derivatives.