Could someone help me with the following conjecture? Thanks a lot!
Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$ with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear inequalities as follows. \begin{equation} \begin{split} \Delta=\{(x_1,\cdots,&x_n)|0\leq x_1\leq n+1,\,0\leq x_2\leq 2(n-1)+1,\,\cdots,\\ &0\leq x_i\leq i(n+1-i)+1,\,\cdots,0\leq x_n\leq n+1,\\ &f_1\geq 0,\,f_2\geq 0,\,\cdots,f_n\geq 0\}, \end{split} \end{equation} where, under the convention $x_0=x_{n+1}=0$, \begin{equation} \begin{split} &f_1(x_1,\cdots,x_n)=2x_1-x_2-x_0=2x_1-x_2,\\ &f_2(x_1,\cdots,x_n)=2x_2-x_3-x_1,\\ &\cdots\\ &f_j(x_1,\cdots,x_n)=2x_j-x_{j+1}-x_{j-1},\\ &\cdots\\ &f_{n-1}(x_1,\cdots,x_n)=2x_{n-1}-x_n-x_{n-2},\\ &f_n(x_1,\cdots,x_n)=2x_n-x_{n+1}-x_{n-1}=2x_n-x_{n-1}.\\ \end{split} \end{equation}
Let $\rho(x_1,\cdots,x_n)$ be the density function over $\Delta$ defined as follows, $$\rho(x_1,\cdots,x_n):=\prod_{1\leq i<j\leq n+1}((x_i-x_{i-1})-(x_j-x_{j-1}))^2.$$ Here we also use the the convention that $x_0=x_{n+1}=0$.
The barycenter $\overline X=(\overline x_1,\cdots,\overline x_n)$ of $\Delta$ with respect to $\rho$ is given by the following formula $$\overline x_i:=\frac{\int_{\Delta}x_i\cdot\rho(x_1,\cdots,x_n)\, dx_1dx_2\cdots dx_n}{\int_{\Delta}\rho(x_1,\cdots,x_n)\, dx_1dx_2\cdots dx_n}$$ where $i=1,\cdots,n$.
Then I would like to know if the following conjecture is true. (I have check the case $n=2,3,4$ by mathematica but do not have any clue how to prove it or disprove it.) $${\bf Conjecture}:\overline x_i>i(n+1-i)\,\,{\rm for\,}i=1,\cdots,n.$$
