We are given an increasing sequence $S_n$ of $n$ distinct real numbers $x_1, x_2, ldots, x_n$, with $x_1=0$ and $x_n=1$. We define the *complementary* distance $D(p,q):=1-d(p,q)$, where $d(p,q):=|x_q-x_p|$. Finally, we define the following functions of a threshold value $tauin(0,1)$:

$$R(tau):=left(sum_{substack{1le i<j<kle n: \ x_i<x_j<tau<x_k}}

D(j,i)right)+

left(sum_{substack{1le i<j<kle n: \ x_i<tau<x_j<x_k}}

D(k,j)right)~,

$$

and

$$M(tau):=sum_{substack{1le i<j<kle n: \ x_i<tau<x_k}}

max{D(j,i),D(k,j)}~.

$$

**Question:** What is (a *sufficiently* tight lower bound for) the minimum value for the ratio $rho^*:=frac{Rleft(tauright)}{Mleft(tauright)}$ over all possible value of $tauin(0,1)$ and all sequences $S_n$ for all $n$? Do we have $rho^*getfrac23$?

*(Can we use random sampling techniques to solve this problem?)*

**Note:** *The difference between this problem and the one described in Combinatorial optimization problem on the sums of differences of real numbers is only the final question.*