{\em   We construct a dense point set of n points in the plane with ne^{\Omega\left({\log n}\right)} halving lines. This improves the bound O (n\log n) of Edelsbrunner, Valtr and Welzl from 1997. We also observe that the upper bound on the maximum number of halving lines of dense point set can be improved to O(n^{7/6 Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in R^d with n^{d-1}e^{\Omega\left({\log n}\right)} halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.
A planar point set of n points is called{\em \gamma-dense if the ratio of the largest and smallest distances among the points is at most \gamma\sqrt{n We construct a dense set of n points in the plane with ne^{\Omega\left({\log n}\right)} halving lines. This improves the bound \Omega (n\log n) of Edelsbrunner, Valtr and Welzl from 1997. Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in R^d with n^{d-1}e^{\Omega\left({\log n}\right)} halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.
A
A planar point set of n points is called
A
\gamma-dense
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if the ratio of the largest and smallest distances among the points is at most \gamma\sqrt{n
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point
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O
\Omega
D
We also observe that the upper bound on the maximum number of halving lines of dense point set can be improved to O(n^{7/6