Bivariate splines on Tohǎneanu partition

In Tohǎneanu (2005), Tohǎneanu introduced a triangulation ΔT with the following property: the space of splines of smoothness r and degree 2r defined on ΔT has a dimension not equal to the lower bound found by Schumaker in Schumaker (1979). In Minàč and Tohǎneanu (2013), it was shown that for d≥2r+1, the dimension of the space of splines on ΔT is equal to the lower bound. We show that this phenomenon can be explained by intrinsic supersmoothness of splines. Moreover, using this supersmoothness we obtain the dimension of the space of splines of smoothness r and degree ≤2r on ΔT.