= Uniform covers theorem
{title2=$|K|^k\leq\prod_{A\in\mathcal A}|K_A|$}
= Uniform cover inequality
{synonym}
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.
Exact covering is essential for unrestricted real volumes. Finite subsets of a Cartesian product satisfy the analogous cardinality inequality, and in that discrete version covering at least $k$ times suffices.
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