Helly-type theorems for polygonal urves
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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...
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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,...
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...
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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...
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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...
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