Separating plane convex sets.

Separating Convex Sets in the Plane

Separating several point sets in the plane

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...

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 $k$ ...

A colorful theorem on transversal lines to plane convex sets

In 1982 (see [1]) Imre Bárány observed that some of the classical theorems in convexity admit interesting and mysterious generalizations which he called “colorful theorems”. For example, the Colorful Helly Theorem says that if a family (repetitions of the same sets are allowed) of compact convex sets in Rk is colored (properly) with k+1 colors and it has the property that any choice of k+1 diff...