On diameter bounds for planar integral point sets in semi-general position

Abstract

A point set M in the Euclidean plane is called a planar integral point set if all the distances between the elements of M are integers, and M is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.

On diameter bounds for planar integral point sets in semi-general position

“…The following result is a key element for bounds presented in [20] and [5]: The height h of the triangle ABC, dropped onto side c, can be found from the formula for its area: S = hc/2, which gives h = 2S/c. To find the area of the triangle, we use the Heron's formula in the following form:…”

“…In [3] the constant (for n ≥ 4) was improved to 0.3457 employing Point Packing in a Square Problem [7,17]. In [6] the approach has been further developed, and the constant has been tightened to 5 11 . Finally, in [5] for IPS M in semi-general position (that is an IPS with no collinear triples) it was proved that diam M ≥ n 5…”

“…In [6] the approach has been further developed, and the constant has been tightened to 5 11 . Finally, in [5] for IPS M in semi-general position (that is an IPS with no collinear triples) it was proved that diam M ≥ n 5…”