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


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




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**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 $\rho(\mathcal{D}^\*)< \frac{20}{21}$?*)




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***Note:*** *This problem can be viewed as the "symmetric version" of question* [Probability distribution optimization problem of distances between points in the interval $[0,1]$](https://mathoverflow.net/q/372341/115803) *and is related to the (discrete) combinatorial problem* [Combinatorial optimization on the sums of differences of real numbers](https://mathoverflow.net/q/365472/115803)



Symmetric distribution optimization problem of distances between points in $[0,1]$
If $\mathcal{D}$ has density of $16/13$ on $[0,13/32]\cup[19/32,1]$, with no support elsewhere, then $\Delta=0.840$, $\Delta'=0.887$, and the ratio is $0.947$. This is less than the $20/21 = 0.952$ from the uniform distribution.


This may not be close to minimal overall, but it's close to minimal for distributions supported uniformly on $[0,u]\cup[1-u,1]$.


