Tolerance in Helly-Type Theorems
نویسندگان
چکیده
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, as well as some other interesting ones.
منابع مشابه
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...
متن کاملDiscrete 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...
متن کاملTransversal 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...
متن کاملHadwiger and Helly-type theorems for disjoint unit spheres
We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...
متن کاملMulti-partite entanglement, communication complexity and quantum Helly-type theorems
We develop protocols for preparing the GHZ state (a pure tri-partite maximally entangled state) and, in general, a pure n-partite maximally entangled state using EPR pairs shared amongst n agents and classical communication between the agents. We observe that the combinatorial arrangement of EPR pairs required for this purpose is like that in Helly-type theorems in geometry. PACS numbers: 03.65...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete & Computational Geometry
دوره 45 شماره
صفحات -
تاریخ انتشار 2011