Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries, and consider the convex optimization problem
$\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$ such that $\sum_{j=1}^k x_j = 1$ and $x_j\ge 0$ for all $j=1...k$.
Assume that $R$ is such that this problem has a unique solution $x$ and that this solution is an interior solution, i.e., $x_j>0$ for all $j=1...k$.
Now assume that the top-left entry of $R$ is raised to some new value $r_{11}'>r_{11}$, giving a new matrix $R'$. Assume that the new solution $x'$ to the above problem is still unique and interior.
Conjecture: $x_1' \ge x_1$.
Task: Prove the conjecture or give a counterexample!