Quantitative combinatorial geometry for concave functions

Sarkar, Sherry; Xue, Alexander; Soberón, Pablo

doi:10.1016/j.jcta.2021.105465

Journal of Combinatorial Theory, Series A

2021

DOI: 10.1016/j.jcta.2021.105465

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Quantitative combinatorial geometry for concave functions

Sherry Sarkar

Alexander Xue

Pablo Soberón

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Cited by 6 publications

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“…For recent applications of Theorem 1.4 to graph theory and to combinatorial geometry see [1] and [8,9,15], respectively. While [1] uses Theorem 1.4 under the stronger assumption that X is d-collapsible, the results in [8,9,15] do require its full version for d-Leray complexes.…”

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confidence: 99%

Relative Leray Numbers via Spectral Sequences

Kalai

Meshulam

2021

Mathematika

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Let F be a fixed field and let X be a simplicial complex on the vertex set V . The Leray number L(X ; F) is the minimal d such that for all i d and S ⊂ V , the induced complex X [S] satisfiesH i (X [S]; F) = 0. Leray numbers play a role in formulating and proving topological Hellytype theorems. For two complexes X, Y on the same vertex set V , define the relative Leray number L Y (X ; F) as the minimal d such thatH i (X [V \ τ ]; F) = 0 for all i d and τ ∈ Y . In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice. §1. Introduction. Let F be a fixed field and let X be a simplicial complex on the vertex set V . All homology and cohomology appearing in the sequel will be with F coefficients. The induced subcomplex of X on a subset S ⊂ V is X [S] = {σ ∈ X : σ ⊂ S}.

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confidence: 99%

Relative Leray Numbers via Spectral Sequences

Kalai

Meshulam

2021

Mathematika

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Isometric and affine copies of a set in volumetric Helly results

Messina

Soberón

2022

Computational Geometry

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Area coverage-based worker recruitment under geo-indistinguishability

et al. 2022

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Quantitative combinatorial geometry for concave functions

Cited by 6 publications

References 41 publications

Relative Leray Numbers via Spectral Sequences

Relative Leray Numbers via Spectral Sequences

Isometric and affine copies of a set in volumetric Helly results

Area coverage-based worker recruitment under geo-indistinguishability

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