| In 1946 Paul Erdo&huml;s posed the following question. What is the maximum number of unit distances that can be determined by a set of n points in the plane? The answer to this question is still out of reach but it has created an important amount of research in Combinatorial Geometry. Here we give an overview of the history and best known bounds for this problem and its natural generalization to higher dimensions. Then we consider the well known restriction of the problem to sets in convex position which inspired the main results of this dissertation. We consider the following problem. Given a strictly convex domain K with smooth boundary, what is the maximum number of unit distances that can be determined by n boundary points of K? We describe sufficient geometric conditions on K under which this number is linear. We prove for example that if K has constant width, or if the curvature of K is less than 2 at every boundary point, then the number described above is at most linear. Other more technical conditions are given. Finally we consider a similar problem: What is the maximum number of (unit) equilateral triangles that can be determined by n points in Rd ? We give several new bounds for this problem, its restriction to sets in convex and general position, and their generalizations to higher dimensions. |
