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## some basics of Prekopa-Leindler inequality

I'm familiar with Prekopa-Leindler as stated in http://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality, for example.

can one say, given $h,f,g$ that satisfy $\forall x,y ,$ $h(tx+(1-t)y) \geq f(x) ^{t} g(y)^{1-t}$ that $$\int e^{h(z)}dz \geq (\int e^{f(x)}dx)^{t} (\int e^{g(y)}dy)^{1-t}$$

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 No. For example, let $f$ be the characteristic function of $[0,1]$, $g=2f$ and $h=\sqrt{2}f$. But the conclusion of course follows if you strengthen the assumption to $h(tx+(1-t)y)\geq tf(x)+(1-t)f(y)$. – Konrad Swanepoel Mar 2 at 23:30