TITLE: What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?
QUESTION [17 upvotes]: My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ subject to the following conditions:
$$\sum_{i=1}^n x_i^2 =\sum_{i=1}^n y_i^2 = 1, \sum_{i=1}^n x_i=\sum_{i=1}^n y_i=0.$$
It seems that $1$ is less or equal to the minimum, but don't have a proof.

REPLY [2 votes]: The lower bound $1$ easily could be obtain. Indeed, for every two zero-sum  vectors $x,y \in \mathbb{R}^n$, we have
$$
\begin{eqnarray}
\left(\sum_{i\leq j} |x_i-x_j||y_i - y_j|\right)^2 
& \geq & \frac 12\sum_{i,j}|x_i -x_j|^2|y_i -y_j|^2 \\ 
& = & \frac 12\sum_{i,j} (x_i^2 -2x_i x_j+x_j^2)(y_i^2-2y_i y_j+y_j^2)\\
& = & n \sum_i (x_i y_i)^2 +2 \Big(\sum_i x_i y_i\Big)^2 + \sum_i x_i^2\sum _i y_i^2 \\ 
& \geq & \|x\|^2 \|y\|^2
\end{eqnarray}
$$