An Extremal Property of the Rellot Triangle

@article{Makeev2003AnEP,
  title={An Extremal Property of the Rellot Triangle},
  author={Vladimir V. Makeev},
  journal={Journal of Mathematical Sciences},
  year={2003},
  volume={113},
  pages={816-817}
}
  • V. Makeev
  • Published 1 March 2003
  • Mathematics
  • Journal of Mathematical Sciences
AbstractLet $$K \subset \mathbb{R}^2 $$ 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. 

References

A Kinematic Formula for Affine Diameters and Affine Medians of a Convex Set
AbstractFor a planar convex set K with C2-smooth boundary, the area of the set of the points lying on a given number of affine diameters of K is estimated. As a corollary, it is proved that the area