Halving lines and measure concentration in the plane

Rom Pinchasi
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09  
Given a set of n points in the plane and a collection of k halving lines of P ℓ1, . . . , ℓ k indexed according to the increasing order of their slopes, we denote by d(ℓj, ℓj+1) the number of points in P that lie above ℓj+1 and below ℓj. We prove an upper bound of 3nk 1/3 for the sum P k−1 j=1 d(ℓj, ℓj+1). We show how this problem is related to the halving lines problem and provide several consequences about measure concentration in R 2 .
doi:10.1145/1542362.1542393 dblp:conf/compgeom/Pinchasi09 fatcat:hnzinbejzrdadcvio5ilyhfune