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 f ^sym (n) 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 .