TITLE: Hölder's Inequality and the Pigeonhole Principle
QUESTION [6 upvotes]: I heard someone in my department claim that Hölder's inequality was just a continuous version of the pigeonhole principle.  It seemed reasonable, but I'm struggling to make their connection precise.
Does anyone know of a precise connection between the two?  Are they related at all, or was my source confused?
[Edit: Originally this post had "Markov's inequality" instead of Hölder's.  I suspect, given my answer below, that my source meant to say Hölder's inequality.]

REPLY [2 votes]: I believe the connection is as follows:
Let $X(\omega)$ be the number of pigeons in the box $\omega$, and let $\mu$ represent the counting measure on the set $\Omega$ of boxes.  Then note that $\mu(\Omega)$ is the number of boxes and $\int X$ is the number of pigeons.  So Hölder's inequality gives: $$\int X \le \mu(\Omega)\|X\|_\infty.$$
Note that $\|X\|_\infty$ is the maximal number of pigeons in any box.  So rearranging, we get the lower bound $\int X /\mu(\Omega) \le \|X\|_\infty$, which says that the quotient of pigeons to boxes gives a lower bound for the number of pigeons in the most populous box, i.e. the pigeonhole principle.
So we can consider Hölder's inequality to be a generalization of the pigeonhole principle.  I don't know if that's a useful way of thinking about it, but at least we found the connection...