On the Reverse Loomis–Whitney Inequality

The present paper deals with the problem of computing (or at least estimating) the LW -number λ(n) , i.e., the supremum of all γ such that for each convex body K in Rⁿ there exists an orthonormal basis { u₁, … , u_n} such that voln(K)n-1i=1nvoln-1(K|ui⊥),where K|ui⊥ denotes the orthogonal projection of K onto the hyperplane ui⊥ perpendicular to u_i. Any such inequality can be regarded as a reverse to the well-known classical Loomis–Whitney inequality. We present various results on such reverse Loomis–Whitney inequalities. In particular, we prove some structural results, give bounds on λ(n) and deal with the problem of actually computing the LW -constant of a rational polytope.