An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets

Abstract

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In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper bound for small values of n, namely, 106≤𝑛≤336 . To obtain this enhancements, we provide lower bounds for the sum of the squares of the degrees of the vertices of a graph related to the halving lines.