Discrete quantitative Helly-type theorems with boxes

Discrete and Lexicographic Helly-Type Theorems

Helly’s theorem says that if every d + 1 elements of a given finite set of convex objects in R have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems—where the common point should belong to an a-priori given set, lexicographic Helly theorems—where the common point should not be lexicographically gr...

Helly-type Theorems for Hollow Axis-aligned Boxes

A hollow axis-aligned box is the boundary of the cartesian product of d compact intervals in Rd. We show that for d ≥ 3, if any 2d of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any 5 of a collection of hollow axis-aligned rectangles in R2 have non-empty intersection, then the whole collection has non-empty ...

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,...

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...