A generalization of Bang's lemma

TL;DR

This paper extends classical geometric lemmas and theorems, including Bang's, Kadets', and the affine plank problem, providing new generalized results and applications to covering problems.

Contribution

It introduces a unified extension of Bang's and Kadets' lemmas for contact pairs and formulates a generalized affine plank problem with specific solutions.

Findings

Generalization of Kadets' theorem

New bounds for contact pairs

Applications to translative and homothetic coverings

Abstract

We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions. In particular, we obtain a generalization of Kadets' theorem. Finally, we give applications to problems regarding translative and homothetic coverings.