Several notes on the circumradius condition

summary:Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1,p}$-error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p=\infty $ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how the case of general $p$ is in fact nothing more than a simple scaling of the standard $O(h)$ estimate under the maximum angle condition, even for higher order interpolation. This allows a direct interpretation of the circumradius estimate and condition in the context of the well established theory of the maximum angle condition