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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.]

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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...

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