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For every Euclidean body $S\subseteq {\mathbb{R}}^n$, there is an axis-parallel box $B$ such thatThe proof minimizes an array of candidate projection volumes subject to the finitely many inequalities from irreducible uniform covers. Tight constraints force the array to factor into its singleton coordinates, which become the side lengths of $B$.

A multiset $\mathcal{C}$ of subsets of $[n]$ is a uniform cover of multiplicity $k$ when each coordinate occurs in exactly $k$ members. The uniform covers theorem states that every Euclidean body $S\subseteq {\mathbb{R}}^n$ satisfieswhere $S_A$ is the coordinate projection of a Euclidean body onto the coordinates in $A$.

We prove it by induction on $n$. Split the cover into ${\mathcal{C}}^{-}$, whose members omit $n$, and ${\mathcal{C}}^{+}$, whose members contain $n$. Exactly $k$ members lie in ${\mathcal{C}}^{+}$. For a last-coordinate value $x$, let $S(x)\subseteq {\mathbb{R}}^{n-1}$ be the corresponding slice. Removing $n$ from the members of ${\mathcal{C}}^{+}$ and retaining the members of ${\mathcal{C}}^{-}$ gives a $k$-uniform cover of $[n-1]$. The inductive hypothesis and Fubini's theorem giveApply Hölder's inequality to the $k$ factors in the integral. Sincewe obtain $|S|\le {\prod }_{A\in \mathcal{C}}|S_A{|}^{1/k}$, which is the result after taking the $k$th power. The one-dimensional base case is immediate.

The Bollobas--Thomason box theorem states that for every body $S\subseteq {\mathbb{R}}^n$ there is an axis-parallel box $B$ such thatAn irreducible uniform cover cannot be decomposed into two smaller uniform covers. There are only finitely many such covers of $[n]$: encode a cover by its multiplicity vector in ${\mathbb{N}}^{2^n}$ and apply the Dickson lemma.

Choose a componentwise minimal positive array $(x_A{)}_{A\subseteq [n]}$ satisfyingfor every irreducible $k$-uniform cover $\mathcal{C}$, together withThe actual projection volumes are feasible by the uniform covers theorem, and finiteness gives a minimal array. Every uniform cover is a disjoint union of irreducible ones, so its cover inequality also holds for this array.

Minimality implies that, for each coordinate $i$, some tight uniform-cover inequality can be chosen whose cover contains the singleton $\{i\}$. Indeed, either such an inequality already blocks decreasing $x_{\{i\}}$, or a tight product inequality $x_A={\prod }_{j\in A}{x}_{\{j\}}$ does; in the latter case take a tight cover containing $A$ and replace that occurrence of $A$ by its singleton coordinates. Let these tight covers be ${\mathcal{C}}_i$, of multiplicities $k_i$, and let $K={\sum }_i{k}_i$. Their multiset union is a $K$-uniform cover. Removing one copy of every singleton leaves a $(K-1)$-uniform cover, so comparison of its cover inequality with the product of all the tight equalities yieldsThe singleton cover gives the reverse inequality, henceFor any $A\subseteq [n]$, the one-uniform cover consisting of $A$ and the singletons $\{i\}$ for $i\notin A$ now gives $x_A\ge {\prod }_{i\in A}{x}_{\{i\}}$. The defining product inequality gives the reverse bound. Thus all these quantities are equal. Taking the side lengths of $B$ to be $x_{\{1\}},\dots ,x_{\{n\}}$ proves the theorem.

Finally suppose that the proper body $S\subseteq {\mathbb{R}}^3$ satisfiesThis is equality in the three-dimensional Loomis--Whitney inequality. In its proof, equality must hold in both applications of Cauchy-Schwarz inequality. Their equality conditions force the three projection indicators to factor through one-dimensional measurable sets $E_1,E_2,E_3$, and forceup to a set of Lebesgue measure zero; this is Equality in the three-dimensional Loomis--Whitney inequality.

Because $S$ is connected, each one-coordinate projection is connected and hence is an interval. Because $S$ is a finite union of positive-volume axis-parallel boxes, a proper difference between $S$ and the product of those three intervals would contain a positive-volume rectangular cell in a common finite subdivision. That would contradict equality up to measure zero. Consequently the equality is exact and

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.