Helly-type theorems for polygonal curves

Helly-type theorems for polygonal urves

Discrete and Lexicographic Helly Theorems and their Relations to LP-Type Problems

Helly’s theorem says that if every d + 1 elements of a given finite set of convex objects in IR 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 g...

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,...

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...

Transversal and Helly-type Theorems in Geometry, Combinatorics and Topology

Helly’s theorem also holds for infinite families of compact convex sets, and has stimulated numerous generalization and variants. Results of the type “if every m members of a family of objects have property P then the entire family has the property P” are called Helly-type theorems. The minimum positive integer m that makes this theorem possible is called the Helly number. Helly-type theorems h...