Separating convex sets by straight lines

Separating Convex Sets in the Plane

On Separating Points by Lines

Given a set P of n points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate n points, picked randomly (and uniformly) in the unit square, is Θ̃(n), where Θ̃ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separa...

Cutting Disjoint Disks by Straight Lines

For k > 0 let f(k) denote the minimum integerf such that, for any family of k pairwise disjoint congruent disks in the plane, there is a direction t~ such that any line having direction ~ intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constants dt, d 2 such that dlx/k.,/~gk~f(k)~d:,x/k~,/~gk. This result has been motiva...

Extracting straight lines by sequential fuzzy clustering

In clustering line segments into a straight line, threshold-based methods such as hierarchical clustering are often used. The line segments comprising a straight line often get misaligned due to noise. Thresholdbased methods have di culty clustering such line segments. A new cluster extraction method is proposed to cope with this problem. This method extracts fuzzy clusters one by one using mat...

Kkm-type Theorems for Products of Simplices and Cutting Sets and Measures by Straight Lines

In this paper a version of Knaster-Kuratowski-Mazurkiewicz theorem for products of simplices is formulated. Some corollaries for measure partition in the plane and cutting families of sets in the plane by lines are given.