Halving lines and measure concentration in the plane

Given a set of n points in the plane and a collection of k halving lines of P ℓ1, . . . , ℓk 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 3nk1/3 for the sum Pk−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 R2 .

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Rom Pinchasi
• Keywords: COMPGEOM 2009, Discrete Geometry, Halving Lines, Show, Upper Bound