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Journal of Mathematical Sciences

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

An Extremal Property of the Rellot Triangle

  • V. V. Makeev
Article
  • 23 Downloads

Abstract

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.

Keywords

Smooth Boundary Constant Width Extremal Property Unique Diameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

REFERENCES

  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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