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Helly’s theorem: New variations and applications (Q2979647)

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English	Helly’s theorem: New variations and applications	scientific article

Statements

scholarly article

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Helly’s theorem: New variations and applications (English)

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Jesús A. De Loera

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Algebraic and Geometric Methods in Discrete Mathematics

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publication date

25 April 2017

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full work available at URL

https://arxiv.org/abs/1508.07606

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zbMATH Keywords

Helly type theorems

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convex sets

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discrete geometry

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optimization

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MaRDI profile type

MaRDI publication profile

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A Helly type theorem for abstract projective geometries

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Very colorful theorems

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A quantitative Doignon-Bell-Scarf theorem

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A Helly-type transversal theorem for \(n\)-dimensional unit balls

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Point Selections and Weak ε-Nets for Convex Hulls

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A colorful theorem on transversal lines to plane convex sets

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Piercing convex sets and the Hadwiger-Debrunner \((p,q)\)-problem

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Bounding the piercing number

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A purely combinatorial proof of the Hadwiger Debrunner \((p,q)\) conjecture

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Transversal numbers for hypergraphs arising in geometry

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An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets

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Helly-type theorems and generalized linear programming

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A short proof of an interesting Helly-type theorem

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Crossing patterns of semi-algebraic sets

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On Maximal $S$-Free Sets and the Helly Number for the Family of $S$-Convex Sets

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A proof of Lovász's theorem on maximal lattice-free sets

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Tight bounds on discrete quantitative Helly numbers

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Transversal numbers over subsets of linear spaces

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A generalization of Caratheodory's theorem

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Minimal Valid Inequalities for Integer Constraints

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Maximal Lattice-Free Convex Sets in Linear Subspaces

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A geometric approach to cut-generating functions

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A Theorem Concerning the Integer Lattice

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Colourful and fractional \((p,q)\)-theorems

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A fractional Helly theorem for boxes

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A Helly-type theorem for semi-monotone sets and monotone maps

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Helly type theorems for the sum of vectors in a normed plane

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Quantitative Helly-Type Theorems

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Helly's Theorem with Volumes

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Helly-type theorems on the homology of the space of transversals

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A fractional Helly theorem for convex lattice sets

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Berge's theorem, fractional Helly, and art galleries

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Jung's theorem for a pair of Minkowski spaces

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Voting in Agreeable Societies

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Colourful Linear Programming and its Relatives

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BRASCAMP–LIEB INEQUALITY AND QUANTITATIVE VERSIONS OF HELLY'S THEOREM

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Starshaped Unions and Nonempty Intersections of Convex Sets in R d

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Helly-type theorems for intersections of sets starshaped via orthogonally convex paths

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A Helly-type theorem for intersections of orthogonally starshaped sets in \(\mathbb R^d\)

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Approximation of convex sets by polytopes

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Optimal guard sets and the Helly property

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The random Tukey depth

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Uncertain convex programs: randomized solutions and confidence levels

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The Scenario Approach to Robust Control Design

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Integer Programming

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On transversals of quasialgebraic families of sets

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Maximal $S$-Free Convex Sets and the Helly Number

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Multinerves and helly numbers of acyclic families

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Helly numbers of acyclic families

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APPROXIMATING CENTER POINTS WITH ITERATIVE RADON POINTS

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Ramsey-type results for semi-algebraic relations

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Lower bounds to Helly numbers of line transversals to disjoint congruent balls

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Helly-type theorems for line transversals to disjoint unit balls

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Helly numbers of polyominoes

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Las Vegas algorithms for linear and integer programming when the dimension is small

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Helly-type theorems for approximate covering

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Helly Type Theorems Derived From Basic Singular Homology

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A Helly type theorem for hypersurfaces

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Helly numbers of algebraic subsets of \(\mathbb{R}^{d}\) and an extension of Doignon's theorem

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Quantitative Tverberg theorems over lattices and other discrete sets

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Quantitative combinatorial geometry for continuous parameters

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Random sampling in computational algebra: Helly numbers and violator spaces

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Convexity in cristallographical lattices

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A Tverberg-type generalization of the Helly number of a convexity space

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Helly dimension of algebraic groups

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An upper-bound theorem for families of convex sets

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Common transversals in the plane: The fractional perspective

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Algorithms - ESA 2003

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The directions of the line segments and of the <i>r</i> ‐dimensional balls on the boundary of a convex body in Euclidean space

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Morris's pigeonhole principle and the Helly theorem for unions of convex sets

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Group actions and Helly's theorem

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A sharp form of Whitney's extension theorem

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The colorful Helly theorem and colorful resolutions of ideals

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Density and regularity theorems for semi-algebraic hypergraphs

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Helly-type theorems for varieties

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A Subexponential Algorithm for Abstract Optimization Problems

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Sampling with removal in LP-type problems

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Cremona convexity, frame convexity and a theorem of Santaló

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Clustering lines in high-dimensional space

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On Components in Some Families of Sets

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Carathéodory, Helly and the others in the max-plus world

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Violator spaces: Structure and algorithms

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Some Discrete Properties of the Space of Line Transversals to Disjoint Balls

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Outline of an algorithm for integer solutions to linear programs

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Bounding Helly numbers via Betti numbers

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On common transversals

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Discrete and lexicographic Helly-type theorems

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Simple stochastic games, parity games, mean payoff games and discounted payoff games are all LP-type problems

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Über eine Variante zum Hellyschen Satz

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Colorful theorems for strong convexity

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A Helly-type theorem for line transversals to disjoint unit balls

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A geometric Hall-type theorem

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Geometric Transversal Theory: T(3)-Families in the Plane

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Points surrounding the origin

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The colored Hadwiger transversal theorem in \(\mathbb{R}^d\)

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Partition numbers for trees and ordered sets

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Line transversals to translates of unit discs

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On a problem by Dol'nikov

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Line transversals to translates of a convex body

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Intersection patterns of convex sets

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Transversals for families of translates of a two-dimensional convex compact set

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Analogues of the central point theorem for families with \(d\)-intersection property in \(\mathbb R^d\)

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Convex sets in the plane with three of every four meeting

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A Problem of Geometry in R n

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A topological colorful Helly theorem

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Leray numbers of projections and a topological Helly-type theorem

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Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers

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Integer Programming with a Fixed Number of Variables

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Lower bounds for a subexponential optimization algorithm

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A Helly-type theorem for unions of convex sets

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Bounded VC-dimension implies a fractional Helly theorem

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Geometric discrepancy. An illustrated guide

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Tolerance in Helly-type theorems

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Transversals, Topology and Colorful Geometric Results

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A new topological Helly theorem and some transversal results

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A proof of Hilbert's Nullstellensatz

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Approximate center points with proofs

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Piercing numbers for balanced and unbalanced families

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A subexponential bound for linear programming

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Dimension gaps between representability and collapsibility

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Approximating Tverberg points in linear time for any fixed dimension

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Proof of a conjecture of Bárány, Katchalski and Pach

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On An Invariant of Plane Regions and Mass Distributions

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Tropical convexity over max-min semiring

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Necessary and sufficient conditions for hyperplane transversals

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A Theorem on General Measure

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Quantitative \((p, q)\) theorems in combinatorial geometry

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Dropping a vertex or a facet from a convex polytope

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An observation on the structure of production sets with indivisibilities

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Small-dimensional linear programming and convex hulls made easy

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Helly-type theorems for the diameter

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A combinatorial bound for linear programming and related problems

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Helly-type theorems for homothets of planar convex curves

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Intersection Patterns of Convex Sets via Simplicial Complexes: A Survey

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Minimum number of elements of representing a set system of given rank

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A Generalization of Radon's Theorem

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Proof of Grünbaum's conjecture on common transversals for translates

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d-collapsing and nerves of families of convex sets

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Identifiers

zbMATH Open document ID

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10.1090/conm/685/13718

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Mathematics Subject Classification ID

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zbMATH DE Number

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Sitelinks

Mathematics(1 entry)

mardi Publication:2979647

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