论文标题
爆炸和kadets不平等的多台面概括
A multi-plank generalization of the Bang and Kadets inequalities
论文作者
论文摘要
如果凸件$ k \ subset \ mathbb {r}^n $由凸面的结合$ c_1，\ ldots，c_n $涵盖，则可以问多个子辅助问题。两个经典结果将宽度的亚加粘性（两个平行的超平面之间的最小距离夹心$ k $）和inradius（$ k $中包含的球的最大半径）：$ c_i $的宽度的总和至少是$ k $的宽度（这是$ k $的宽度（这是$ k $）（这是$ plank theore $ ume $ ume $ sump thig thigi $ sump aump yis th guger and th gungi $ cum aump）至少$ k $的inradius（这是由于Vladimir Kadets造成的）。 我们调整了这些结果的现有证据，以证明某些广义非凸“多板”的覆盖率定理。这种方法的一种推论是一个不平等的家族，在Bang定理和Kadets定理之间进行了插值。其他推论还包括让人联想到达文波特的结果 - 诸如以下问题，例如：如果在飞机上$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n unitk a inradius a $ nradius atradius at inradius at inradius at rac { \sinπ/m} $。
If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, 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 $C_i$ is at least the width of $K$ (this is the plank theorem of Thoger Bang), and the sum of the inradii of the $C_i$ 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 $\frac{\sin π/m}{N + \sin π/m}$.
