Фигури с постоянна ширина

This is a popular-science article based on known results from the literature. It surveys bodies of constant width, from the Reuleaux triangle and Reuleaux polygons in the plane to the Reuleaux tetrahedron and the Meissner bodies in space. We derive the perimeter and the area of the Reuleaux triangle and of Reuleaux polygons, show why the Reuleaux tetrahedron does not have constant width, and explain how rounding three of its edges repairs this defect. An exact expression for the volume of a Meissner body is derived from its surface area and Blaschke's relation between volume and surface area. The open Bonnesen–Fenchel conjecture on the minimal volume is also discussed.