  Various results are proved giving lower bounds for the $m$th intrinsic volume
$V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in
terms of the $m$th intrinsic volumes of its projections on the coordinate
hyperplanes (or its intersections with the coordinate hyperplanes). The bounds
are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively)
forms of the Loomis-Whitney inequality and versions of it that apply to
intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean
width, the inequality obtained confirms a conjecture of Betke and McMullen made
in 1983.
