A multi-plank generalization of the Bang and Kadets inequalities

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