What is the minimum of this double sum expression?

Question

Given $n$ non-negative values. Their sum is $k$.

$$ \sum_{i=1}^n x_i = k $$

The double sum expression is defined as:

$$ \sum_{i=1}^n\big((\sum_{j=i}^n x_j)x_i\big)$$

I think that the expression reaches a minimum when $x_i = k/n$. It is true for $n=2$. Does it hold for all $n$? And how to prove it?

Thanks!

Answer 1

you can rewrite it as $$ \frac 12 \left( (\sum x_i)^2 + \sum x_i^2 \right) = \frac 12 \left( k^2 + \sum x_i^2 \right) $$ and to prove that the minimum is reached for $x_i = k/n$ it is just enough to apply AM-GM or Cauchy-Schwartz