Journal of Mathematical Sciences

, Volume 113, Issue 6, pp 816–817 | Cite as

An Extremal Property of the Rellot Triangle

  • V. V. Makeev


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.


Smooth Boundary Constant Width Extremal Property Unique Diameter 
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  1. 1.
    I. M. Yaglom and V. G. Boltyanskii, Convex Sets [in Russian], Gostekhizdat, Moscow (1951).Google Scholar
  2. 2.
    V. V. Makeev, “A kinematic formula for affinne diameters and affine medians of a convex set,” Zap. Nauchn. Semin. POMI, 280, 234–238(2001).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • V. V. Makeev
    • 1
  1. 1.St.Petersburg State UniversityRussia

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