Box theorem (${\prod }_i{b}_i=|K|,{\prod }_{i\in A}{b}_i\le |K_A|$)

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 ${\sum }_{i\in A}{x}_i\le \log |K_A|$. 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.

Set $x_i=\log b_i$ for candidate axis-parallel box side lengths. Its coordinate-projection constraints are the displayed linear program. The dual feasible weights are fractional uniform covers. Its optimal extreme point is rational, so the uniform covers theorem forces its objective to be at least $\log |S|$. Weight one on the full coordinate set attains that value. Strong duality therefore supplies box side lengths with volume $|S|$ and no larger coordinate projections.