The inequality $2(x_0y_0+x_1y_1)-(x_0+x_1)(y_0+y_1)<0$ can be easily reduced to the equivalent form $(x_1-x_0)(y_1-y_0)<0$, which allows to find the solutions the first inequality immediately (by comparing x_0,y_0 with x_1,y_1) .
Is there a similar trick for the inequality $n\sum_{i=0}^n x_i y_i-(\sum_{i=0}^n x_i)(\sum_{i=0}^n y_i)<0$ when $n>2$?
I am particularly interested in the case $n=16$.
We have $$2n\sum_{i=1}^nx_iy_i-2\left (\sum_{i=1}^nx_i\right ) \left (\sum_{i=1}^ny_j\right )\\ =\sum_{i=1}^n\sum_{j=1}^n[x_iy_i+x_jy_j] - \sum_{i=1}^n\sum_{j=1}^n[x_iy_j+x_jy_i]\\ =\sum_{i=1}^n\sum_{j=1}^n(x_i-x_j)(y_i-y_j)=2\sum_{i<j}(x_i-x_j)(y_i-y_j)$$ In particular, if both sequences are simultaneously increasing or simultaneously decreasing then the outcome is nonnegative. If one is increasing while the other one decreasing the outcome is nonpositive.