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 ℝd. 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 ℱ = {ℐh}h → 0 of nested face-to-face partitions ℐh. For d = 2, we prove that this family is strongly regular; i.e., there exists a constant C > 0 such that meas T ≥ Ch2 for all triangles T ∈ ℐh and all triangulations ℐh ∈ ℱ. In particular, the well-known minimum angle condition is valid.