A proof of convergence and an error bound for the method of bisection in $\textbf {R}^{n}$

Abstract:

Let $S = \langle {X_0},..,{X_m}\rangle$ be an m-simplex in ${{\mathbf {R}}^n}$. We define "bisection" of S as follows. We find the longest edge $\langle {X_i},{X_j}\rangle$ of S, calculate its midpoint $M = ({X_i} + {X_j})/2$, and define two new m-simplexes ${S_1}$ and ${S_2}$ by replacing ${X_i}$ by M or ${X_j}$ by M. Suppose we bisect ${S_1}$ and ${S_2}$, and continue the process for p iterations. It is shown that the diameters of the resulting Simplexes are no greater then ${(\sqrt 3 /2)^{\left \lfloor {p/m} \right \rfloor }}$ times the diameter of the original simplex, where $\left \lfloor {p/m} \right \rfloor$ is the largest integer less than or equal to $p/m$.