I am wondering if the following generalization of van der Waerden's conjecture is true.


Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T\_i$. Then $per(A) \geq T\_1\ldots T\_n \frac{n!}{n^n}$. This obviously implies van der Waerden's conjecture. I can check it by hand for 2x2 matrices, and I did not have the patience to try larger examples (so it may be false for some easy example). I couldn't modify Gurvits's proof to work either.



A generalization of van der Waerden's conjecture
The conjecture is false. Here is a counterexample.


\begin{equation\*}
 A = \begin{bmatrix}
 \tfrac18 & \tfrac4{15} & \tfrac1{10}\\
 \tfrac18 & \tfrac4{15} & \tfrac1{10}\\
 \tfrac68 & \tfrac7{15} & \tfrac8{10}
 \end{bmatrix}.
\end{equation\*}


For this matrix, $\text{per}(A)=\frac{21}{200}=0.105$. The row sums $T\_1,\ldots,T\_3=\left\{\frac{59}{120},\frac{59}{120},\frac{121}{60}\right\}$, so that $T\_1T\_2T\_3 n!/n^n = \frac{421201}{3888000} \approx 0.108334...$, which is greater than $\frac{21}{200} = 0.105$.


