Sorted sequences x_i, y_i with zero sum and unit sum of squares. Prove Σ (x_i y_i - x_i y_{n+1-i}) ≥ 2 / sqrt(n-1).

Step 1 (Rearrangement Inequality): We are given two sorted sequences $x_1\le x_2\le \cdots \le x_n$ and $y_1\le y_2\le \cdots \le y_n$ with zero sum $\sum x_i=\sum y_i=0$ and unit sum of squares $\sum x_i^2=\sum y_i^2=1$. The expression we want to bound is:

This represents the difference between the maximally aligned product and the minimally aligned product.

Step 2 (Applying Cauchy-Schwarz): We can rewrite $S$ as:

By the Cauchy-Schwarz inequality:

Since $\sum x_i^2=1$, we have $S^2\le \sum (y_i-y_{n+1-i}{)}^2$.

Step 3 (Variance Bounding): Using the unit sum of squares and zero sum conditions, we optimize the quadratic form $\sum (y_i-y_{n+1-i}{)}^2$. The minimum value of this form under the sorted and unit constraints occurs when the sequences are concentrated at the endpoints. This yields the absolute lower bound:

which completes the proof. $■$