TITLE: Geometric interpretation of Young's inequality
QUESTION [24 upvotes]: Is there a geometric interpretation of Young's inequality, $$ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$$ with $\dfrac{1}{p}+\dfrac{1}{q} = 1$? 
My attempt is to say that $ab$ could be the surface of a rectangle, and that we could also say that:
$\dfrac{a^{p}}{p}=\displaystyle \int_{0}^{a}x^{p-1}dx$, 
but them I'm stuck.

REPLY [6 votes]: For positives $a$, $b$, $p$ and $q$ we'll rewrite our inequality in the following form.
$$\ln\left(\frac{a^p}{p}+\frac{b^q}{q}\right)\geq\frac{1}{p}\ln{a^p}+\frac{1}{q}\ln{a^q},$$
which is just Jensen for the concave function $\ln$, which is geometry, of course.