A Mélange of Diameter Helly-Type Theorems

Tolerance in Helly-Type Theorems

In this paper we introduce the notion of tolerance in connection with Helly type theorems and prove, using the Erdős-Gallai theorem, that any Helly type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly type theorems, such as Caratheodory’s and Tverberg’s theorems,...

Some Computational Aspects of Helly-type Theorems

In this paper, we prove that, for a given positive number d, if every n + 1 of a collection of compact convex sets in IE contain a set of width d (a set of constant width d, respectively) simultaneously, then all members of this collection contain a set of constant width d1 simultaneously, where d1 = d/ √ n if n is odd and d1 = d √ n+ 2/ (n+ 1) if n is even (d1 = 2d− d √ 2n/(n+ 1), respectively...

Helly-type theorems for polygonal urves

Helly-type theorems for polygonal curves

We prove the following intersection and covering Helly-type theorems for boundaries of convex polygons in the plane. • Let S be a set of points in the plane. Let n¿ 4. If any 2n+2 points of S can be covered by the boundary of a convex n-gon, then S can be covered by the boundary of a convex n-gon. The value of 2n+ 2 is best possible in general. If n= 3, 2n+ 2 can be reduced to 7. • Let S be a 4...

Helly-Type Theorems in Property Testing

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If S is a set of n points in R, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly’s theorem, by taking ...