= Three projection areas of a volume-one body
{title2=$abc\geq1$}
Positive numbers $a,b,c$ are the areas of the three coordinate projections of a volume-one <Euclidean body> exactly when $abc\geq1$. Necessity is the three-dimensional <Loomis--Whitney inequality>. For sufficiency take box lengths $x=\sqrt{ac/b}$, $y=\sqrt{ab/c}$ and $z=\sqrt{bc/a}$, and retain the union of the three slabs with first, second or third normalized coordinate at most $t$. All three projections are full rectangles, while volume is $\sqrt{abc}[1-(1-t)^3]$. Choosing $t=1-(1-1/\sqrt{abc})^{1/3}$ gives volume one. The slabs have positive thickness, so the construction uses a body with interior rather than zero-volume additions.
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