Given parameters $w_1, \ldots, w_n > 0$ with $\sum_{j=1}^n w_j = 1$ ( $n\ge 2$), define function $$ f(x_1, \ldots, x_{n-1})= \sum_{j=1}^{n} \frac{x_j}{1-x_j} \cdot \sum_{j=1}^n \frac{w_j}{1-w_j} - \sum_{j=1}^n \frac{x_j}{1-w_j} \cdot \sum_{j=1}^n \frac{w_j}{1-x_j} $$ with $$x_n = 1-\sum_{j=1}^{n-1}x_j$$ for $x_1, \dots, x_{n-1} \in (0,1)$. For example, for $n=3$ the function is $$ f(x_1, x_2)= \left ( \frac{x_1}{1-x_1} + \frac{x_2}{1-x_2}+ \frac{1- x_1-x_2}{x_1+x_2}\right) \left ( \frac{w_1}{1-w_1} + \frac{w_2}{1-w_2}+ \frac{w_3}{1-w_3}\right)- \left ( \frac{w_1}{1-x_1} + \frac{w_2}{1-x_2}+ \frac{w_3}{x_1+x_2}\right) \left ( \frac{x_1}{1-w_1} + \frac{x_2}{1-w_2}+ \frac{1- x_1-x_2}{1-w_3}\right)$$ Working on a conjecture in this MO question, for $n= 3, 4, 5$, I want to numerically check whether the Hessian matrix of $f(x_1, \ldots, x_{n-1})$ is positive semi-definite over the hypercube $(0,1)^{n-1}$ for any parameters $w_1, \ldots, w_n > 0$ with $\sum_{j=1}^n w_j = 1$. I could manage the case $n=3$ using a desmos graph, which shows the answer is yes. Could you help me efficiently use Mathematica to handle the two remaining cases $n=4,5$? I found a related question discussing how one can check whether a matrix is positive semi-definite by Mathematica.

Thanks! It was very helpful. In fact, the determinant of the Hessian matrix may have sharp drops near some of the extreme points of the hypercube, which cannot be detected by checking the Desmos figure.

I think your plot of the determinant misses the behaviour near the origin. Just calculate the determinant at `x=1/100`, `y=1/100`, for example.

Thank you Gianluca for your kind consideration! However, it is not consistent with what I see from my method at the point. In this link, I depicted the two diagonal elements and the determinant of the Hessian matrix. You may see that all the three quantities are highly positive at the point you obtained (see the intersection of the green and orange lines). The reason may be that $x_3$ is used instead of $1-x_1-x_2$, or something else.

Checking whether Hessian matrix of a function is positive semidefinite on a hypercube