The number of unit-area triangles in the plane: Theme and variations

We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if $S$ consists of points on three lines, the number of unit-area triangles that $S$ spans can be $\Omega(n^2)$, for any triple of lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\em convex grid} of the form $A\times B$, where $A$, $B$ are {\em convex} sets of $n^{1/2}$ real numbers each (i.e., the sequences of differences of consecutive elements of $A$ and of $B$ are both strictly increasing), then $S$ determines $O(n^{31/14})$ unit-area triangles

Configurations of Lines in 3-Space and Rigidity of Planar Structures

Let L be a sequence (l_1,l_2,...,l_n) of n lines in C^3. We define the intersection graph G_L=([n],E) of L, where [n]:={1,..., n}, and with {i,j} in E if and only if ineq j and the corresponding lines l_i and l_j intersect, or are parallel (or coincide). For a graph G=([n],E), we say that a sequence L is a realization of G if G subset G_L. One of the main results of this paper is to provide a combinatorial characterization of graphs G=([n],E) that have the following property: For every generic realization L of G, that consists of n pairwise distinct lines, we have G_L=K_n, in which case the lines of L are either all concurrent or all coplanar. The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes-Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs

Sets with few distinct distances do not have heavy lines

Let $P$ be a set of $n$ points in the plane that determines at most $n/5$ distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span