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Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that $$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A 1)^3. $$

This problem was asked by @Pavel Gubkin in math stackexchange without an answer.

It is related to Problem 10210 in AMM 1992 No. 3:

(a) Let $f$ be a continuous non-negative real-valued function defined on the squares $[0, 1]^2$. Show that $$\int_0^1\int_0^1\int_0^1\int_0^1f(x_1, y_1)f(x_2, y_1)f(x_1, y_2)\,dx_1\,dx_2\,dy_1\,dy_2$$ $$\ge \left(\int_0^1\int_0^1f(x,y)\,dx\,dy\right)^3.$$ See: The American Mathematical Monthly Vol. 99, No. 3 (Mar., 1992), pp. 265-266.

I tried to seek the official solution, but failed.

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  • $\begingroup$ Do you know how to prove No 3 (a)? $\endgroup$
    – mathworker21
    Commented Dec 5 at 14:19
  • $\begingroup$ @mathworker21 No. Any solutions are welcome. $\endgroup$
    – River Li
    Commented Dec 5 at 14:29
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    $\begingroup$ See mathoverflow.net/questions/189222/… for the integral inequality. The discrete one might be a particular case, not sure (no time to check). $\endgroup$
    – darij grinberg
    Commented Dec 5 at 15:04
  • $\begingroup$ @darijgrinberg Thanks. Yes, the discrete case is addressed in the accepted answer for that question. $\endgroup$
    – River Li
    Commented Dec 5 at 15:17

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