Convex Polygons are Self-Coverable

TL;DR

This paper introduces the concept of self-coverability for geometric families, proves that homothets of convex polygons are self-coverable, and derives new coloring results related to cover-decomposability.

Contribution

It establishes self-coverability for homothets of convex polygons and connects this to coloring problems, advancing understanding of cover-decomposability in geometric families.

Findings

Homothets of convex polygons are self-coverable.

New coloring results for point sets with respect to geometric families.

Connections established between self-coverability and cover-decomposability.

Abstract

We introduce a new notion for geometric families called self-coverability and show that homothets of convex polygons are self-coverable. As a corollary, we obtain several results about coloring point sets such that any member of the family with many points contains all colors. This is dual (and in some cases equivalent) to the much investigated cover-decomposability problem.