Journal ArticleDOI
A Note on Smaller Fractional Helly Numbers
TLDR
It is shown that F, as well as P, for any given set P∩P∣F∈F, have fractional Helly number at most k, which improves the known bounds for fractionalHelly numbers of many families.Abstract:
Let $${\mathcal {F}}$$F be a family of geometric objects in $${\mathbb {R}}^d$$Rd such that the complexity (number of faces of all dimensions on the boundary) of the union of any m of them is $$o(m^k)$$o(mk). We show that $${\mathcal {F}}$$F, as well as $$\{F \cap P \mid F \in {\mathcal {F}}\}$${F?P?F?F} for any given set $$P \in {\mathbb {R}}^d$$P?Rd, have fractional Helly number at most k. This improves the known bounds for fractional Helly numbers of many families.read more
Citations
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Journal ArticleDOI
The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg
TL;DR: Five discrete results are discussed: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg fromCombinatorial geometry, which explore their connections and emphasize their broad impact in application areas.
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The discrete yet ubiquitous theorems of Carath\'eodory, Helly, Sperner, Tucker, and Tverberg
TL;DR: In this paper, the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath-eodory, Helly, and Tverberg are discussed.
Proceedings ArticleDOI
On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers
TL;DR: A polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property is introduced.
Journal ArticleDOI
Improved bounds on the Hadwiger–Debrunner numbers
TL;DR: For families in R2 with union complexity below a specific quadratic bound, it was shown in this article that for any ϵ > 0, there exists a p0 = p0(ϵ) such that for every p ≥ p0 and for every q ≥ log p, the family H{D_d}(p,d) = \tilde O(p + {(p/q)^d})
Posted Content
Improved bounds on the Hadwiger-Debrunner numbers
TL;DR: In this article, a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying the Hadwiger-Debrunner conjecture was proposed.
References
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Helly, Radon, and Carathéodory Type Theorems
TL;DR: In this paper, the authors discuss applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory.
Journal ArticleDOI
Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem
Noga Alon,Daniel J. Kleitman +1 more
TL;DR: In this paper, it was shown that for every p ⩾ q ⊾ d + 1 there is a c = c(p, q, d) < ∞ such that for any family J of compact, convex sets in road, there is at most c points in road that intersects each member of J.
Journal ArticleDOI
A Problem of Geometry in R n
M. Katchalski,A. Liu +1 more
TL;DR: The main result in this article is that if almost all the (n + l)-subfamilies of Ti have nonempty intersection, then fS has a subfamily with non-empty intersection containing almost all of the sets in T.AssTRAcr.