Who is the author?

I'm not sure.

Ok, no problem.

Note that $(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2=2(p^2+q^2+r^2+s^2-(pq+qr+rs+sp))=2(a^2+b^2+c^2+d^2)-2(p+r)(q+s)$, and so we need to maximize $(p+r)(q+s)$. We have $(a+b)(c+d)>(a+c)(b+d)\iff ac+bd>ab+cd\iff (a-d)(c-b)>0$ which is true and also $(a+b)(c+d)<(a+d)(b+c)\iff ad+bc<ab+cd\iff (a-c)(d-b)<0$ which is true, and so $(a+d)(b+c)$ gives the maximal product. Therefore, to minimize the initial expression we can take the permutations in $\{(a,b,d,c),(a,c,d,b),(b,a,c,d),(b,d,c,a),(c,d,b,a),(c,a,b,d),(d,b,a,c),(d,c,a,b)\}$