= Solution
The <uniform cover inequality> states that for a positive-volume body $K\subset\mathbb R^n$ and a multiset $\mathcal A$ in which each coordinate occurs exactly $k$ times, $\boxed{|K|^k\leq\prod_{A\in\mathcal A}|K_A|}$, where $K_A$ is its coordinate projection and volumes use the corresponding dimensions.
To deduce the <box theorem>, maximize $\sum_i x_i$ subject to $\sum_{i\in A}x_i\leq\log|K_A|$ for every nonempty $A\subseteq[n]$. This <linear program> is feasible and bounded above by the full-set constraint. Its dual minimizes $\sum_A\lambda_A\log|K_A|$ over nonnegative weights with $\sum_{A\ni i}\lambda_A=1$. For rational weights, clearing denominators gives an exact uniform cover, so the <uniform cover inequality> bounds that dual objective below by $\log|K|$. The same holds for all feasible weights by density of rational points in the rational constraint polyhedron. <Strong duality> makes the primal optimum at least $\log|K|$, while the full-set constraint makes it at most that value. An optimal solution therefore gives $b_i=e^{x_i}>0$ with
$$
\boxed{\prod_i b_i=|K|,\qquad\prod_{i\in A}b_i\leq|K_A|\quad\text{for every }A.}
$$
The axis-aligned box with these side lengths has the required volume and projection volumes.
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