We are given an increasing sequence $S$ of positive real numbers $x_1, x_2, \ldots, x_n$, such that $$x_{i+2}-x_{i+1} \ge c\,(x_{i+1}-x_i)$$ for all $i=1, 2, \ldots n-2$, where $c\ge 1$ is constant. Each number $x_i\in S$ is associated with a positive integer weight $w_i$ for all $i=1, 2, \ldots, n$. Let $W$ be the sequence formed by these weights.

Finally, let $$A=\sum_{1\le i < j < k\le n} w_i w_j w_k(x_j-x_i)~,$$ and $$B=\sum_{1\le i < j < k\le n} w_i w_j w_k\min(x_k-x_j, x_j-x_i)~.$$

**Question**: What is the minimum value for $c$ such that, *for all $S$ and* $W$, we have $A \le 2B~$? (*I am also interested in tight upper bounds*).

* Conjectures*:

*I believe that $c\ge \frac{3}{2}$ is a sufficient condition to obtain $A \le 2B$ as required (I do not know whether it is also necessary). Furthermore, I think that the worst case w.r.t. $W$ occurs when $w_{n}\gg n$ and $w_{n-1} \gg n$, while all the other weights $w_1, w_2, \ldots, w_{n-2}$ are equal to $1$. Finally, I also conjecture that, for any $\alpha\ge 2$, $c\ge 1+\frac{1}{\alpha}$ is a sufficient condition to obtain $A \le \alpha B$.*