An extremal property of the Rellot triangle

Let $K\subset\mathbb R^2$ be a planar set of unit constant width with piecewise $C^2$-smooth boundary. Then the area of the set of the points belonging to $\ge3$ diameters of $K$ is $\le\sqrt3/4$, and the area of the set of the points belonging to a unique diameter of $K$ is $\ge(2\pi-3\sqrt3)/4$. In both cases, an equality is attained only if $K$ is the Rellot triangle.