The below listed 3 equations are given:
$$A = 1-\sum_{i=1}^n x_i^2,$$ $$B = 1-\sum_{i=1}^n y_i^2,$$ and $$C = 1-\sum_{i=1}^n x_i y_i$$
with $x_i, y_i \in [0,1]$ and $\sum_{i=1}^n x_i = \sum_{i=1}^n y_i = 1$.
Is it possible to write $C$ as a function of $A$ and $B$, i.e. as $C(A,B)$?
Every help is appreciated! Many thanks in advance!
No. Consider $x_i = y_i = 1$ vs $x_i = -y_i = 1$.
In both cases, $A = B = 1 - n$. In the first case $C = 1 - n$ and in the second $C = 1 + n$. Therefore knowing $A$ and $B$ does not tell you $C$.