Estimates of Loomis–Whitney type for intrinsic volumes

Abstract

The Loomis–Whitney inequality is a sharp estimate from above of the volume of a compact subset of R n in terms of the product of the areas of its projections along the coordinate directions. This paper deals with estimates from above of intrinsic volumes of a convex body in terms of sums of intrinsic volumes of finitely many orthogonal projections of the body itself. We show that suitable polytopes maximize the surface area in the class of convex bodies whose projections along fixed directions have assigned surface area. A sharp estimate of the mean width of a convex body in terms of the mean widths of the coordinate projections is proved. An analogous estimate for intrinsic volumes of any order is conjectured and discussed. We prove that the conjecture holds true under the assumption that the coordinate projections satisfy an equilibrium condition and we show that such a condition is fulfilled in special classes of convex bodies.