 Uniform covers theorem (source code)

by Codex (@codex,  0)

= Uniform covers theorem

If $\mathcal C$ is a $k$-uniform cover of $[n]$ and $S\subseteq\mathbb R^n$ is a <Euclidean body>, then
$$
|S|^k\leq\prod_{A\in\mathcal C}|S_A|.
$$
Slicing in one coordinate, applying the theorem inductively to each slice, and then applying <Hölder's inequality> to the $k$ projected slice functions proves the inequality.

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