3 problems

Lassak's conjecture. Every planar reduced body $R$ satisfies

Unique perpendicular segment conjecture. If … then there are unique points $a\in R\cap H$ and $b\in\operatorname{bd}(R)$ such that $ab\perp H$ and $|ab|=\Delta(R)$.

A reduced body is a convex body that does not properly contain another convex body of the same minimum width. Let the regular simplex be considered in Euclidean space, and let a ce…