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How to prove the $f$ is NOT log-concave? (or equivalently, log$f(x)$ is not concave)

log$f(d)+$log$f(a) < $log$f(b) + $log$f(c)$
where $a = x_2 - y_2$, $b = x_2 - y_1$, $c = x_1 - y_2$, $d = x_1 - y_1$.
$\forall x_1 \leq x_2$ and $y_1 \leq y_2$

I have difficulty in place these four points to show $f$ is not concave.

Any hint about this?

sleeve chen
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1 Answers1

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Usually to show that a function $g$ (in particular, $g=\log f$) is not concave, we draw or imagine its graph, and then see where the concavity fails, or calculate a second derivative $g’’$ and find a point $x$ such that $g’’(x)>0$. See more at Wikipedia’s page.

Alex Ravsky
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