I've been reading this paper and in remark 2.1. it is mentioned that the estimate (9) from Def 2.1. $$\|\nabla \varphi \|_{L^\infty (K)}\leq \frac{C}{diam \Omega}$$ follows from the regularity of the mesh, in the case of piecewise linear hat function on triangles.
I am looking for a proper proof of this! Can anyone help or give a hint?
Start by proofing the estimate on the unit triangle with vertices in $(0,0),(1,0),(0,1)$ and the hat function $$f(x,y)=-x-y+1.$$ Then use the inverse of the transformation $$ \begin{pmatrix} x_2-x_1 & x_3-x_1 \\ y_2-y_1 & y_3-y_1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}x_1\\y_1\end{pmatrix},$$ where $x_1,..,x_3,y_1,..,y_3$ are the coordinates of the vertices of any triangle, to find the hat function on that triangle. Now use $x_2-x_1\leq diam\Omega$ and similar inequalities and shape regularity that tells you: $$\exists c\geq 0:\ \frac{1}{sup\{\text{radius of inscribed circle}\}}\leq \frac{c}{diam\Omega}$$