3 problems

Wetzel's conjecture. The triangle $T$ covers every unit arc in the plane.

Gerriets–Poole's conjecture. A region is a cage if and only if it can accommodate every polygonal worm.

Let $\mathcal{F}$ be the family of all arcs of unit length, and let a circular sector have angle $30^\circ$ and unit radius. The 30-degree circular-sector covering conjecture. This…