Geodesic
An extremal property of the Rellot triangle
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 5, Tome 267 (2000), pp. 152-155

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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. 
@article{ZNSL_2000_267_a9,
     author = {V. V. Makeev},
     title = {An extremal property of the {Rellot} triangle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {152--155},
     publisher = {mathdoc},
     volume = {267},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_267_a9/}
}
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V. V. Makeev. An extremal property of the Rellot triangle. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 5, Tome 267 (2000), pp. 152-155. http://geodesic.mathdoc.fr/item/ZNSL_2000_267_a9/