Do we have an explicit expression of this function? Let $\Delta^{n-1} \equiv \{(p_1, \ldots, p_n) \, |\, p_i \geq 0, \sum_i p_i = 1\}$ be the $n-1$ dimensional simplex. Define $f : \Delta^{n-1} \times \Delta^{n-1} \rightarrow \Delta^{n-1}$ such that $$ f(p, q) = \Big(\frac{p_1 q_1}{\sum_i p_i q_i}, \ldots, \frac{p_n q_n}{\sum_i p_i q_i}\Big).$$ Suppose $h : \Delta^{n-1} \rightarrow \mathbb{R}$ be a function such that $h(f(p, q)) = \sum_i p_i q_i$. Do we have an explicit expression of $h$?
Thank you!

No, we don't because there is no such function for $n\ge 2$. Suppose
$$
\exists h:\Delta^{n-1}\to \Bbb R\ \ : \ \ h(f(p,q))=\sum_{i=1}^n p_iq_i.
$$ For $p=q=(\frac{1}{2},\frac{1}{2},0,0,\ldots,0)$, it holds $f(p,q)=(\frac{1}{2},\frac{1}{2},0,0,\ldots,0)$ and $$h\left(\frac{1}{2},\frac{1}{2},0,0,\ldots,0\right)=h(f(p,q))=p\cdot q =\frac{1}{2}.$$ However,
For $p'=(\frac{1}{3},\frac{2}{3},0,0,\ldots,0)$ and $q'=(\frac{2}{3},\frac{1}{3},0,0,\ldots,0),$ it holds $f(p',q')=(\frac{1}{2},\frac{1}{2},0,0,\ldots,0)$ and $$h\left(\frac{1}{2},\frac{1}{2},0,0,\ldots,0\right)=h(f(p',q'))=p'\cdot q' =\frac{4}{9}.$$ This leads to a contradiction.
