Alternative Equation Formats

$${\displaystyle d^{\ast <!--\; \ast \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}(x_i,x_j)=\{\begin{array}{lcl}\frac{1}{2}(\frac{1}{f(x_i)}+\frac{1}{f(x_j)})& & \text{if }d(x_i,x_j)\le <!--\; \le \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>max(r_k(x_i),r_k(x_j))\\ \mathrm{\infty <!--\; \infty \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}& & \text{otherwise}\end{array}}$$

$${\displaystyle d^{\ast <!--\; \ast \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}(x_i,x_j)=\{\begin{array}{lcl}\frac{1}{2}(\frac{1}{f(x_i)}+\frac{1}{f(x_j)})& & \text{if }d(x_i,x_j)\le <!--\; \le \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>max(r_k(x_i),r_k(x_j))\\ \mathrm{\infty <!--\; \infty \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}& & \text{otherwise}\end{array}}$$

\[ d^*(x_ i , x_ j) = \left\{ \begin{array}{lcl} \frac{1}{2} \left( \frac{1}{f(x_ i)} + \frac{1}{f(x_ j)} \right) & & \mbox{if } d(x_ i , x_ j) \leq \max (r_ k(x_ i) , r_ k(x_ j)) \\[0.10in] \infty & & \mbox{otherwise} \end{array} \right. \]