Is there a geometric intuition for Hölder's inequality?
I am referring to $||fg||_1 \le ||f||_p ||f||_q $, when $\frac{1}{p}+\frac{1}{q}=1$.
For $p=q=2$ this is just the Cauchy-Schwarz inequality, for which I have geometric intution: The projection of a vector along a direction shortens its length.
My question is if there is a similar geometric interpretation for Hölder's inequality. I am aware of the scaling argument, which shows the inequality can only hold when $\frac{1}{p}+\frac{1}{q}=1$; but why should we expect this to be true when the $p,q$ are conjugates? Perhaps there is some physical interpretation?
Note that I am looking for intuition, not necessarily a formal proof. Hölder's inequality can be proved using Young's inequality, for which a beautiful intuition is given here.
In my perspective, even though this gives intuition to a component in the proof of Hölder's inequality, this does not really give an intuition for the inequality itself.
(But maybe I am wrong? does the actual integration have "true content" in it, or is Hölder really nothing but Young's inequality in disguise? Part of the confusion is that the intuition for Young's inequality is based on integration, so if we only rely on that, the intuition for Hölder should be some sort of "double integration"... )
To see that the geometric intuition of Young's and Hölder's inequalities are somewhat different, we can look at $p=q=2$:
In that case, Young's inequality is just the standard AM-GM inequality for two variables. This inequality can be interpreted geometrically. Although here one can also view this as "projection only shortens", the scenario is a little different than the one in the Cauchy-Schwarz inequality. (At least the the reasons behind the "equality cases" seem slightly different to me).