### Numbers of Weights of Convex Quadrilaterals in Weighted Point Sets

Toshinori Sakai, Satoshi Matsumoto

#### Abstract

Let ℘n 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,...,n}. For P∈℘n and a polygon Q with vertices in P, we define the weight of Q as the sum of the weights of its vertices and denote by Wk(P) the set of weights of convex k-gons with vertices in P∈℘n. Let fk(n) = minP∈℘n |Wk(P)|. It is known that n-5 ≤ f4(n) ≤ 2n-9 for n≥7. In this paper, we show that f4(n)≥ 4n/3-7.

#### Keywords

point set, weight, convex quadrilateral

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DOI: http://dx.doi.org/10.19184/ijc.2024.8.1.1

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