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It was previously shown that $$H\ge cG,\tag{1}$$ where $c:=1/14334$, $$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$ and $X,Y,Z$ are independent random variables with the same log-concave density.

It is clear that the constant factor $c=1/14334$ is not optimal.

Question: What is the best (that is, the largest) possible constant factor $c$ in (1)?

The following might turn out to be of help:

Conjecture: The best possible value of $c$ is $1/9$, attained when $X$ has an exponential distribution (which would thus represent the least favorable case).

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  • $\begingroup$ I am looking for a good answer as well! Thanks for "crosslink" my question! $\endgroup$
    – Fei Cao
    Jun 30, 2021 at 17:46

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