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.]