On a generalization of the plank problem

TL;DR

This paper generalizes the plank problem by analyzing how angular domains covering a smaller circle within a larger concentric circle relate to the view angle from points in the larger circle.

Contribution

It introduces a generalized framework for the plank problem involving concentric circles and angular domains, extending classical results to a new geometric setting.

Findings

Sum of angles of covering domains exceeds the view angle from the larger circle

Provides a geometric inequality relating covering angles and view angles

Extends classical plank problem to concentric circle configuration

Abstract

Let two concentric circles $k$ and $K$ be given on the plane with radii $r$ and $R$, $r<R$. Assume that the disc bounded by $k$ is covered by angular domains whose vertices are within $K$. Then the sum of the angles of these angular domains is greater than or equal to the view angle of $k$ from an arbitrary point of $K$.