An Extremal Property of the Rellot Triangle

Abstract

Let $K \subset \mathbb{R}^2$ be a planar set having unit constant width and piecewise $C^2$-smooth boundary. Then the area of the set of the points belonging to at least three diameters of K is at most $\sqrt 3 /4$, and the area of the set of the points belonging to a unique diameter of K is at least $(2\pi - 3\sqrt 3 )/4$. In both cases, an equality is attained only if K is the Rellot triangle. Bibliography: 2 titles.