= Maximum-volume open rectangular box at fixed surface area
{title2=$V_{\max}=(S/3)^{3/2}/2$}
For a lidless <rectangular parallelepiped> with positive dimensions $l,w,h$ and prescribed external <surface area> $S=lw+2lh+2wh$, the largest <volume> is attained at $l=w=2h=\sqrt{S/3}$. The <arithmetic-geometric mean inequality> applied to $lw,2lh,2wh$ gives
$$
4(lwh)^2\leq(S/3)^3,
$$
with equality exactly when these three face-area terms are equal. This proves both the claimed global bound and its attainable equality case. <Lagrange multipliers> produce the same dimensions from the interior stationarity equations. The <weighted open-box minimization> is the corresponding fixed-volume formulation with general positive face weights.
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