Some Old and New Problems in Combinatorial Geometry

Usually we the number the points Let x l ,x , . . .,xn be n distinct points in a metric space . will restrict ourselves to the plane . Denote by D(x , . . .,x ) of distinct distances determined by x , . . .,x . Assume lthat n are in r-dimensional space . Denote byl n f (n)= min D(x, . . .,x), r xl, . . .,x l n n I conjectured more than 40 years ago that f (n)> c n/(log n) 2 . The lattice points show that this if true is best possible . In this paper we discuss problems related to the conjecture and other questions related to this parameter . I wrote many papers with this or similar titles, and will try to avoid repetitions as much as possible. Let xl,x2, . . .,xn be n distinct points in a metric space. Usually we will restrict ourselves to the plane . Denote by D(xl, . . .,x„) the number of distinct distances determined by x l ,. . .,xn . Assume that the points are in r-dimensional space . Denote by fr(n) = min D(xl, . . .,xn) . X 1, . . .Xn I conjectured more than 40 years ago that (1) f2(n) > c1n/(log n)1i2 . I offer five hundred dollars for a proof or disproof of (1) . The lattice points show that *The Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary .