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Discrete & Computational Geometry

, Volume 28, Issue 4, pp 467–473 | Cite as

The Unit Distance Problem for Centrally Symmetric Convex Polygons

  • Ábrego
  • Fernández-Merchant
Article

Abstract

Let f(n) be the maximum number of unit distances determined by the vertices of a convex n -gon. Erdos and Moser conjectured that this function is linear. Supporting this conjecture we prove that f sym (n)
$$\sim$$
2n where f sym (n) is the restriction of f(n) to centrally symmetric convex n -gons. We also present two applications of this result. Given a strictly convex domain K with smooth boundary, if f K (n) denotes the maximum number of unit segments spanned by n points in the boundary of K , then f K (n)=O(n) whenever K is centrally symmetric or has width >1.

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Ábrego
    • 1
  • Fernández-Merchant
    • 1
  1. 1.Department of Mathematics, California State University, Northridge, Northridge, CA 91330-8313, USA bernardo.abrego@csun.edu, silvia.fernandez@csun.eduUSA

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