An extremal property of the Reuleaux triangle (Q1404777)

Let \(K\) be a planar set having unit constant width and piecewise \(C^2\)-smooth boundary. The author considers the subset of \(K\) determined by all points in \(K\) lying on at least three diameters of \(K\). Using differentiability properties and integral geometry techniques the author obtains an upper bound of the area of this subset. This upper bound is attained if and only if \(K\) is the Reuleaux triangle. An analogous result is proved for a lower bound of the area of the subset of \(K\) determined by all the points in \(K\) lying on a unique diameter of \(K\).