Geodesic diameter of bodies of constant width

Abstract

Let Φ a three-dimensional body of constant width B. Then the geodesic diameter G of the surface of Φ is estimated via B from above and from below. It is proved that $$ G \leqslant \frac{\pi }{2}B $$ , where an equality occurs and only if Φ is a body of revolution. Bibliography: 3 titles.