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) $\sim$ 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 . [PUBLICATION ABSTRACT]