Abstract:
Theorems of the form "if every m objects from a set has a certain property P, then
the entire set has the property P" are known as Helly-type theorems, after Eduard
Helly's theorem of 1923. Many transversal theorems on collections of bodies in Rn
have been stated and proved in this Helly-type style. In this paper, we primarily
focus on the planar case of transversals providing support to a collection of convex
bodies.
Previous authors have shown that given a sufficiently large family of pairwise
disjoint convex planar bodies with the property that any three bodies share a support
line, the entire family shares at least one support line. Unfortunately, sufficiently
large does not provide an exact number of bodies necessary to ensure that if any
three bodies have the support property, then the entire family does.
Using what is called a (special) set-system of words and letters to model these
disjoint convex planar bodies, the existing literature shows us that the minimum
number of bodies necessary to ensure the aforementioned support property is no
more than 143. In this paper we do not provide the exact number, but rather a
generalized framework for which this planar case is a special case.
Consistent with earlier authors, we too use a set-system of words and letters.
Within our framework, these systems are known as k-systems and special k-systems.
We provide a few Helly-type results on both k-systems and special k-systems.
