In the definition of the Riemann integral we choose a partition of an interval as follows: $\displaystyle \pi = \{a = x_{0} < x_{1} < \cdots < x_{n} = b \} $ and set $\displaystyle \Delta x_i = x_i - x_{i-1} $.

Then we define the mesh as $\displaystyle ||\pi|| = \max\limits_{1 \leq i \leq n} \Delta x_i $. So is this the maximum distance $\displaystyle \Delta x_i $?

Why not define it as $\displaystyle ||\pi|| = \min\limits_{1 \leq i \leq n} \Delta x_i $?