Logarithmic linear program for the box theorem ($\max {\sum }_i{x}_i:{\sum }_{i\in A}{x}_i\le \log |S_A|$)

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.