The unit distance problem for centrally symmetric convex polygons

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 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 width > 1.