How does combinatorial geometry study packing and covering problems involving geometric figures?

In combinatorial geometry, packing and covering problems focus on the economical arrangement of geometric figures within a given space or relative to other figures.12

Packing Problems

Packing involves placing figures of a specific shape or size without overlap, aiming for the greatest possible efficiency or density. A classic example is the study of how many equal spheres can touch a central sphere of the same size; while it is intuitive that 12 spheres can surround one in three-dimensional space, proving that 13 cannot fit was a significant historical challenge. Another famous instance is Kepler’s conjecture, which concerns the densest possible packing of spheres in space, a problem that remained unsolved for centuries until it was proved in 1998.12

Covering Problems

Covering problems are the conceptual inverse, seeking the most economical way to arrange figures so that their union contains another given figure. A notable example is the search for a "universal cover"—a shape of minimal area that can cover any set of a given diameter.1

These problems often rely on extremal properties, where researchers identify an optimal or "extreme" configuration to solve broader geometric questions.3