LAUSR

If a convex body in $\mathbb{R}^n$ is covered by the union of convex bodies, multiple subadditivity questions can be asked. The subadditivity of the width is the subject of the celebrated plank theorem of Th. Bang, whereas the subadditivity of the inradius is due to V. 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 Davenport's potato problem, and certain inequalities on the relative width.