Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, i.e., $x\le y\le z$, . Let $a=y-x$ and $b=z-y$. We define

$$\Delta=1-\mathbb{E}\left[\frac{b}{a+b}\cdot a+\frac{a}{a+b}\cdot b\right]=1-\mathbb{E}\left[\frac{2ab}{a+b}\right]$$

and

$$\Delta'=1-\mathbb{E}\left[\min(a,b)\right]~.$$

**Question:** What is the minimum value of the ratio $\rho(\mathcal{D})=\frac{\Delta}{\Delta'}$ over all probability distributions $\mathcal{D}$?
(*When $\mathcal{D}$ is uniform in $[0,1]$, we have $\rho=\frac{20}{21}$. Is there a distribution $\mathcal{D}^*$ such that $\rho(\mathcal{D}^*)< \frac{20}{21}$?*)

**Note:***This problem can be viewed as the "symmetric version" of question* Probability distribution optimization problem of distances between points in the interval $[0,1]$ *and is related to the (discrete) combinatorial problem* Combinatorial optimization on the sums of differences of real numbers

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