Separating convex sets in the plane

Separating Convex Sets in the Plane

Separating several point sets in the plane

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

Convex Hulls of Sets in the Plane

The convex hull of a set of points is the smallest convex set containing the points. The convex hull is a fundamental concept in mathematics and computational geometry. Other problems can be reduced to finding the convex hull – for example, halfspace intersection, Delaunay triangulation, Voronoi diagram, etc. An algorithm known as Graham scan [6] achieves an O(n log n) running time. This algori...

The combinatorial encoding of disjoint convex sets in the plane

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2...