Bollobas--Thomason box theorem
= Bollobas--Thomason box theorem
{c}
For every <Euclidean body> $S\subseteq\mathbb R^n$, there is an <axis-parallel box> $B$ such that
$$
|B|=|S|,
\qquad
|B_A|\leq|S_A|\quad(A\subseteq[n]).
$$
The 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 $B$.