Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Repeated angles in the plane and related problems
Journal of Combinatorial Theory Series A
Extremal problems in combinatorial geometry
Handbook of combinatorics (vol. 1)
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications
ICALP '89 Proceedings of the 16th International Colloquium on Automata, Languages and Programming
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Intersection reverse sequences and geometric applications
Journal of Combinatorial Theory Series A
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
Combinatorics, Probability and Computing
The Minimum Number of Distinct Areas of Triangles Determined by a Set of $n$ Points in the Plane
SIAM Journal on Discrete Mathematics
Distinct Triangle Areas in a Planar Point Set
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
An improved bound on the number of unit area triangles
Proceedings of the twenty-fifth annual symposium on Computational geometry
Farthest segments and extremal triangles spanned by points in R3
Information Processing Letters
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The study of extremal problems on triangle areas was initiated in a series of papers by Erdös and Purdy in the early 1970s. Here we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an O(n44/19) = O(n2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first improvement of the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases: We show: (i) For points in convex position, there exist n-element point sets that span Ω(n log n) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2/3(n2 -- n); there exist n-element point sets (for arbitrarily large n) that span (6/π2 -- o(1))n2 minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span (n log n) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an O(n17/7β(n)) = O(n2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is another classical result of Erdös and Purdy from 1971. We further show, for point sets in3-space: (i) The number of minimum nonzero area triangles is at most n2 + O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span (n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε 0. (iii) Every set of n points, not all on a line, determines at least (n2/3/β(n)) triangles of distinct areas, which share a common side.