Strong regularity of a family of face-to-face partitions generated by the longest-edge bisection algorithm

We examine the longest-edge bisection algorithm that chooses for bisection the longest edge in a given face-to-face simplicial partition of a bounded polytopic domain in $\mathbb R^d$. Dividing this edge at its midpoint, we define a locally refined partition of all simplices that surround this edge. Repeating this process, we obtain a family $\mathscr F=\{\mathscr T_h\}_{h\to0}$ of nested face-to-face partitions $\mathscr T_h$. For $d=2$, we prove that this family is strongly regular; i.e., there exists a constant $C>0$ such that $\operatorname{meas}T\ge Ch^2$ for all triangles $T\in\mathscr T_h$ and all triangulations $\mathscr T_h\in\mathscr F$. In particular, the well-known minimum angle condition is valid.