The Unit Distance Problem for Centrally Symmetric Convex Polygons

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 fsym (n) 2n where fsym (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 fK (n) denotes the maximum number of unit segments spanned by n points in the boundary of K, then fK (n) = O (n) whenever K is centrally symmetric or has

• Type: Journal
• Year: 2002
• Where: DCG
• Authors: Bernardo M. Ábrego, Silvia Fernández-Merchant
• Keywords: Convex N-gons, DCG 2002, Maximum Number, Unit Distances