We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L2-norm, that is ‖f − g‖2 = M (f − g)2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching. 1 Introduction and problem statement

To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have come out of the blue: usually the paper I refer to adds the last step to earlier ideas. Since this is an extended abstract (of a nonexistent paper), I will be rather brief, or sometimes completely silent, about the history, with apologies to the unmentioned giants on whose shoulders the authors I do mention have been standing. 1 A careful reader may notice that together with these great results, I will also advertise some smaller results of mine. • Larry Guth and Nets Hawk Katz [16] completed a bold project of György Elekes (whose previous stage is reported in [10]) and obtained a neartight bound for the Erdős distinct distances problem: they proved that every n points in the plane determine at least Ω(n / log n) distinct distances. This almost matches the best known upper bound of O(n / √ √ √ log n), attained for the n × n grid. Their proof and some related results and methods constitute the main topic of this note, and will be discussed later. • János Pach and Gábor Tardos [27] found tight lower bounds for the size of ε-nets for geometric set systems. 2 It has been known for a long time