Numbers of Weights of Convex Quadrilaterals in Weighted Point Sets

Abstract

<p style="text-align: justify;">Let ℘<sub>n</sub> denote the family of sets of points in general position in the plane each of which is assigned a different number, called a weight, in {1,2,...,<em>n</em>}. For <em>P</em>∈℘<sub>n</sub> and a polygon Q with vertices in <em>P</em>, we define the weight of <em>Q</em> as the sum of the weights of its vertices and denote by <em>W</em><sub>k</sub>(<em>P</em>) the set of weights of convex <em>k</em>-gons with vertices in <em>P</em>∈℘<sub>n</sub>. Let <em>f</em><sub>k</sub>(<em>n</em>) = min<sub><em>P</em>∈℘<sub>n</sub></sub> |<em>W</em><sub>k</sub>(<em>P</em>)|. It is known that <em>n</em>-5 ≤ <em>f</em><sub>4</sub>(<em>n</em>) ≤ 2<em>n</em>-9 for <em>n</em>≥7. In this paper, we show that <em>f</em><sub>4</sub>(<em>n</em>)≥ 4<em>n</em>/3-7.</p>