= Box theorem
{title2=$\prod_i b_i=|K|,\quad\prod_{i\in A}b_i\leq|K_A|$}
= Bollobas--Thomason box theorem
{c}
{synonym}
A <Euclidean body> has an <axis-parallel box> of the same volume with every coordinate-projection volume no larger than its own. In logarithmic coordinates the box side lengths solve a <linear program> with constraints $\sum_{i\in A}x_i\leq\log|K_A|$. Its dual uses nonnegative exact fractional covers; the <uniform cover inequality> bounds every dual objective below by $\log|K|$, proving existence by <strong duality>.
An alternative 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 the box.
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