3,463 research outputs found
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 of
points in the plane is , improving the earlier bound
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 consists of points on three lines, the number of
unit-area triangles that spans can be , for any triple of
lines (it is always in this case). (ii) We show that if is a {\em
convex grid} of the form , where , are {\em convex} sets of
real numbers each (i.e., the sequences of differences of consecutive
elements of and of are both strictly increasing), then determines
unit-area triangles
Sets with few distinct distances do not have heavy lines
Let be a set of points in the plane that determines at most
distinct distances. We show that no line can contain more than points of . 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
- …