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Pré-Publication, Document De Travail Année : 2011

Transversals to the convex hulls of all k-set of discrete subsets of R^n

Résumé

Let k,d,\lambda\ge 1 be integers with d\ge 1. What is the maximum positive integer n such that every set of n points in R^d has the property that the convex hulls of all k-sets have a transversal (d-\lambda)-plane? What is the minimum positive integer n such that every set of n points in general position in Rd has the property that the convex hulls of all k-sets do not have a transversal (d-\lmabda)-plane? In this paper, we investigate these two questions. We de ne a special Kneser hypergraph and, by using some topological results and the well-known -Helly property, we relate our second question to the chromatic number of such hypergraphs. Moreover, we establish a connection (when \lmabda = 1) with Kneser's conjecture, rst proved by Lovasz. Finally, we prove a discrete at center theorem.
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Dates et versions

hal-00812962 , version 1 (18-04-2013)

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  • HAL Id : hal-00812962 , version 1

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Jorge Arocha, Javier Bracho, Luis Montejano, Jorge Ramirez Alfonsin. Transversals to the convex hulls of all k-set of discrete subsets of R^n. 2011. ⟨hal-00812962⟩
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