Conjectured maximum number of minimum-area triangles in N no-three-in-line lattice points

Friday, February 12, 2016
Xuanlin Li, Tennessee Academy of Science, Chattanooga, TN
In the project, a new case concerning extremal problems in lattice triangle areas is proposed: Given a set of N (N≥3) lattice points in the plane and no any three points are allowed to be collinear, what is the maximum number of triangles that have smallest area among the C(N,3) triangles formed? It was proved that for all sizes of minimum area, the answer was above 2N+4⌊N/8⌋-8 and below 2N(N-1)/3. Then, a construction of N=2 (2≤k≤8) lattice points was given to estimate the true answer, which could also be applied to all sizes of minimum area. The conjectured lower bound was  ⌊88(N-1)/27⌋, which is linear as N approaches infinity. The conjectured upper bound was O(N ln N), which is slightly above linear.