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I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

Immediately after Lev Borisov's sketch of a proof idea below, Lev Boris, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039

As announced in my first post, the prize winner is Lev Borisov.

I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

Immediately after Lev Borisov's sketch of a proof idea below, Lev Borisov, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039 ${}{}{}$

As announced in my first post, the prize winner is Lev Borisov.

I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

Immediately after Lev Borisov's sketch of a proof idea below, Lev Boris, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039

As announced in my first post, the prize winner is Lev Borisov.

I'm searching for the first proof (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture