Suppose I am given two sets of real numbers $\{a_i\}_{i=1}^N$ and $\{w_i\}_{i=1}^N$ with $w_i>0$. I am trying to find the maximum of the expression
$\left\lvert \sum_i a_i \left(\frac{w_i s_i}{\sum_j w_j s_j}-s_i\right) \right\rvert$
over a simplex $\Delta:= \{\{s_i\}: \sum_i s_i=1, s_i\ge 0\}\subset \mathbb{R}^N$. I suspect the answer is
$\max_{i,j} \left\lvert(a_i-a_j) \frac{\sqrt{w_i}-\sqrt{w_j}}{\sqrt{w_i}+\sqrt{w_j}} \right\rvert\;,$
which is obtained by having $s_i=\frac{\sqrt{w_j}}{\sqrt{w_i}+\sqrt{w_j}}$ and $s_j=\frac{\sqrt{w_i}}{\sqrt{w_i}+\sqrt{w_j}}$ for some $i$ and $j$ and having the remaining $s_k=0, k\ne i, k\ne j$. How can I show it is true, or if it is not true, how do I solve this problem? Thanks.