 Uniform covers theorem

If $\mathcal{C}$ is a $k$-uniform cover of $[n]$ and $S\subseteq {\mathbb{R}}^n$ is a Euclidean body, thenSlicing 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.