ON SOME METRIC AND COMBINATORIAL GEOMETRIC PROBLEMS

I have published many papers on this and related topics [l]. Important progress has been made over the last few years on many of these problems and I will give a short review of some of these at the end of this paper and also state there some of the remaining problems, but first of all I will state some new problems. Usually we will restrict ourselves to the plane though many interesting questions can be posed in higher dimensions and even on the line (though the problems on the line are almost entirely of number theoretic and combinatorial character); also I almost entirely ignore our numerous problems and results with George Purdy since we plan to write both a survey paper and a book on these questions, but enough of idle talk and let us see some action. Let x1, x2, . . . , x, be n distinct points in the plane, denote by D(xl, . . . , x,) the number of distinct distances determined by xi, . . . , x,. Put