A multi-plank generalization of the Bang and Kadets inequalities

Abstract

If a convex body K ⊂ ℝn is covered by the union of convex bodies C1, …, CN, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich K) and the inradius (the largest radius of a ball contained in K): the sum of the widths of the Ci is at least the width of K (this is the plank theorem of Thøger Bang), and the sum of the inradii of the Ci is at least the inradius of K (this is due to Vladimir Kadets).We adapt the existing proofs of these results to prove a theorem on coverings by certain generalized non-convex “multi-planks”. One corollary of this approach is a family of inequalities interpolating between Bang’s theorem and Kadets’s theorem. Other corollaries include results reminiscent of the Davenport-Alexander problem, such as the following: if an m-slice pizza cutter (that is, the union of m equiangular rays in the plane with the same endpoint) in applied N times to the unit disk, then there will be a piece of the partition of inradius at least \({{\sin \pi /m} \over {N + \sin \pi /m}}\).