A generalization of Bang's lemma

A multi-plank generalization of the Bang and Kadets inequalities

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 the…

Around Sperner's lemma

We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also shown that the main theorem implies Tucker's lemma and some other theorems.

Sawayama-Thébault定理的推广的证明

Proof of the Generalization of the Sawayama-Thébault Theorem

本文证明了2016年提出的两个猜想，展示了推广定理在Laguerre几何中的应用，为问题提供了解决框架。