Look at the expression $$ f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1. $$ The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "cyclic correlations" of consecutive variables. Then you can check that $f$ is minimized for the values $x_1=x_2=x_3=1$.

Now look at $$ g(x_1,x_2,x_3,x_4) = x_1^2+x_2^2+x_3^2+x_4^2+x_1x_2+x_2x_3+x_3x_4+x_4x_1, $$ under the assumption that $x_1+x_2+x_3+x_4=4$. Again, this is minimized by $x_1 = x_2 = x_3 = x_4=1$.

A similar thing happens if I add "second-order cyclic correlations": Let $$ h(x_1,x_2,x_3,x_4) = x_1^2+x_2^2+x_3^2+x_4^2+x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_1x_3+x_2x_4+x_3x_1+x_4x_2, $$ again under the assumption that $x_1+x_2+x_3+x_4=4$. This is also minimized for the values $x_1=x_2=x_3=x_4=1$.

Is there a simple explanation for this? Is there a simple argument showing that the same will happen for, say, 12 variables and correlations of order up to 3?