Reverse and dual Loomis-Whitney-type inequalities

Various results are proved giving lower bounds for the mth intrinsic volume Vm(K), m = 1,...,n−1, of a compact convex set K in Rn, in terms of the mth 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 V1(K), which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983.