research-article

Multinerves and helly numbers of acyclic families

Authors:

�ric Colin de Verdi�re,

Gr�gory Ginot,

Xavier GoaocAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 209 - 218

https://doi.org/10.1145/2261250.2261282

Published: 17 June 2012 Publication History

Abstract

The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.

References

[1]

N. Alon and G. Kalai. Bounding the piercing number. Discrete & Computational Geometry, 13:245--256, 1995.

Digital Library

[2]

N. Amenta. Helly-type theorems and generalized linear programming. Discrete & Computational Geometry, 12:241--261, 1994.

Digital Library

[3]

N. Amenta. A new proof of an interesting Helly-type theorem. Discrete & Computational Geometry, 15:423--427, 1996.

Digital Library

[4]

D. Attali and H. Edelsbrunner. Inclusion-exclusion formulas from independent complexes. Discrete & Computational Geometry, 37:59--77, 2007.

Digital Library

[5]

D. Attali, A. Lieutier, and D. Salinas. Efficient data structure for representing and simplifying simplicial complexes in high dimensions. In Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SoCG '11, pages 501--509, 2011.

Digital Library

[6]

D. Attali, A. Lieutier, and D. Salinas. Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. In Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SoCG '11, pages 491--500, 2011.

Digital Library

[7]

C. Borcea, X. Goaoc, and S. Petitjean. Line transversals to disjoint balls. Discrete & Computational Geometry, 1--3:158--173, 2008.

Digital Library

[8]

K. Borsuk. On the imbedding of systems of compacta in simplicial complexes. Fundamenta Mathematicae, 35:217--234, 1948.

[9]

R. Bott and L. W. Tu. Differential forms in algebraic topology, volume 82 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982.

[10]

G. E. Bredon. Sheaf theory, volume 170 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.

[11]

F. Chazal and S. Y. Oudot. Towards persistence-based reconstruction in Euclidean spaces. In Proceedings of the 24th Annual Symposium on Computational Geometry, SoCG '08, pages 232--241, 2008.

Digital Library

[12]

O. Cheong, X. Goaoc, and A. Holmsen. Lower bounds to Helly numbers of line transversals to disjoint congruent balls. Israel Journal of Mathematics, 2012. To appear.

[13]

O. Cheong, X. Goaoc, A. Holmsen, and S. Petitjean. Hadwiger and Helly-type theorems for disjoint unit spheres. Discrete & Computational Geometry, 1--3:194--212, 2008.

Digital Library

[14]

O. Cheong, X. Goaoc, and H.-S. Na. Geometric permutations of disjoint unit spheres. Computational Geometry: Theory & Applications, 30:253--270, 2005.

Digital Library

[15]

�. Colin de Verdi�re, G. Ginot, and X. Goaoc. Helly numbers of acyclic families. arXiv:1101.6006, 2011.

[16]

L. Danzer, B. Gr�nbaum, and V. Klee. Helly's theorem and its relatives. In V. Klee, editor, Convexity, Proceedings of Symposia in Pure Mathematics, pages 101--180. AMS, 1963.

[17]

V. de Silva and G. Carlsson. Topological estimation using witness complexes. IEEE Symposium on Point-Based Graphics, pages 157--166, 2004.

Digital Library

[18]

H. Debrunner. Helly type theorems derived from basic singular homology. American Mathematical Monthly, 77:375--380, 1970.

[19]

J. Eckhoff. Helly, Radon and Caratheodory type theorems. In J. E. Goodman and J. O'Rourke, editors, Handbook of Convex Geometry, pages 389--448. North Holland, 1993.

[20]

J. Eckhoff and K.-P. Nischke. Morris's pigeonhole principle and the Helly theorem for unions of convex sets. Bulletin of the London Mathematical Society, 41:577--588, 2009.

[21]

H. Edelsbrunner and N. R. Shah. Triangulating topological spaces. International Journal of Computational Geometry and Applications, 7(4):365--378, 1997.

[22]

B. Farb. Group actions and Helly's theorem. Advances in Mathematics, 222:1574--1588, 2009.

[23]

R. Godement. Topologie alg�brique et th�orie des faisceaux. Hermann, Paris, 1973. Troisi�me �dition revue et corrig�e, Publications de l'Institut de Math�matique de l'Universit� de Strasbourg, XIII, Actualit�s Scientifiques et Industrielles, No. 1252.

[24]

P. G. Goerss and J. F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhauser Verlag, Basel, 1999.

[25]

V. Goryunov and D. Mond. Vanishing cohomology of singularities of mappings. Compositio Mathematica, 89(1):45--80, 1993.

[26]

M. J. Greenberg. Lectures on Algebraic Topology. Benjamin, Reading, MA, 1967.

[27]

B. Gr�nbaum and T. Motzkin. On components in some families of sets. Proceedings of the American Mathematical Society, 12:607--613, 1961.

[28]

A. Hatcher. Algebraic topology. Cambridge University Press, 2002. Available at http://www.math.cornell.edu/ hatcher/.

[29]

E. Helly. �ber Mengen konvexer K�rper mit gemeinschaftlichen Punkten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175--176, 1923.

[30]

E. Helly. �ber Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatshefte f�r Mathematik und Physik, 37:281--302, 1930.

[31]

A. Holmsen. Recent progress on line transversals to families of translated ovals. In J. E. Goodman, J. Pach, and R. Pollack, editors, Computational Geometry - Twenty Years Later, pages 283--298. AMS, 2008.

[32]

G. Kalai and R. Meshulam. Intersections of Leray complexes and regularity of monomial ideals. Journal of Combinatorial Theory, Series A, 113:1586--1592, 2006.

Digital Library

[33]

G. Kalai and R. Meshulam. Leray numbers of projections and a topological Helly-type theorem. Journal of Topology, 1:551--556, 2008.

[34]

M. Kashiwara and P. Schapira. Sheaves on manifolds, volume 292 of Grundlehren der Mathematischen Wissenschaften {Fundamental Principles of Mathematical Sciences}. Springer-Verlag, Berlin, 1990.

[35]

J. Matou\v sek. Using the Borsuk-Ulam Theorem. Springer-Verlag, 2003.

[36]

J. Matousek. A Helly-type theorem for unions of convex sets. Discrete & Computational Geometry, 18:1--12, 1997.

[37]

J. P. May. Simplicial objects in algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original.

[38]

H. Morris. Two pigeon hole principles and unions of convexly disjoint sets. Phd thesis, California Institute of Technology, 1973.

[39]

L. Santal�. Un theorema sobre conjuntos de paralelepipedos de aristas paralelas. Publ. Inst. Mat. Univ. Nat. Litoral, 2:49--60, 1940.

[40]

S. Smorodinsky, J. S. B. Mitchell, and M. Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in Rd. Discrete & Computational Geometry, 23:247--259, 2000.

[41]

E. H. Spanier. Algebraic topology. McGraw-Hill Book Co., New York, 1966.

[42]

R. Stanley. f-vectors and $h$-vectors of simplicial posets. J. Pure and Applied Algebra, 71:319--331, 1991.

[43]

H. Tverberg. Proof of Gr�nbaum's conjecture on common transversals for translates. Discrete & Computational Geometry, 4:191--203, 1989.

Digital Library

[44]

R. Wenger. Helly-type theorems and geometric transversals. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete & Computational Geometry, chapter 4, pages 73--96. CRC Press LLC, Boca Raton, FL, 2nd edition, 2004.

Digital Library

[45]

Y. Zhou and S. Suri. Geometric permutations of balls with bounded size disparity. Computational Geometry: Theory & Applications, 26:3--20, 2003.

Digital Library

Cited By

Yan LMasood TSridharamurthy RRasheed FNatarajan VHotz IWang B(2021)Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific VisualizationComputer Graphics Forum10.1111/cgf.1433140:3(599-633)Online publication date: 29-Jun-2021
https://doi.org/10.1111/cgf.14331
Carrière MOudot S(2018)Structure and Stability of the One-Dimensional MapperFoundations of Computational Mathematics10.1007/s10208-017-9370-z18:6(1333-1396)Online publication date: 1-Dec-2018
https://dl.acm.org/doi/10.1007/s10208-017-9370-z
Curto C(2016)What can topology tell us about the neural code?Bulletin of the American Mathematical Society10.1090/bull/155454:1(63-78)Online publication date: 27-Sep-2016
https://doi.org/10.1090/bull/1554
Show More Cited By

Index Terms

Multinerves and helly numbers of acyclic families
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics

Recommendations

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in źd. In particular, we show that a family of nź2d disjoint unit balls in źd has a line transversal if, for some ordering ź of the balls, any subfamily of 2d balls admits a line ...
A Mélange of Diameter Helly-Type Theorems

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove ...
Carath�odory, Helly, and Radon Numbers for Sublattice and Related Convexities

The Carath�odory, Helly, and Radon numbers are three main invariants in convexity theory. These invariants have been determined, exactly or approximately, for a number of different convexity structures. We consider convexity structures defined by the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright � 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2012

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

SoCG '12

Sponsor:

SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
91
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 15 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Yan LMasood TSridharamurthy RRasheed FNatarajan VHotz IWang B(2021)Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific VisualizationComputer Graphics Forum10.1111/cgf.1433140:3(599-633)Online publication date: 29-Jun-2021
https://doi.org/10.1111/cgf.14331
Carri�re MOudot S(2018)Structure and Stability of the One-Dimensional MapperFoundations of Computational Mathematics10.1007/s10208-017-9370-z18:6(1333-1396)Online publication date: 1-Dec-2018
https://dl.acm.org/doi/10.1007/s10208-017-9370-z
Curto C(2016)What can topology tell us about the neural code?Bulletin of the American Mathematical Society10.1090/bull/155454:1(63-78)Online publication date: 27-Sep-2016
https://doi.org/10.1090/bull/1554
Adamaszek MAdams HFrick FPeterson CPrevite-Johnson C(2016)Nerve Complexes of Circular ArcsDiscrete & Computational Geometry10.1007/s00454-016-9803-556:2(251-273)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1007/s00454-016-9803-5
Halperin DKerber MShaharabani D(2015)The Offset Filtration of Convex ObjectsAlgorithms - ESA 201510.1007/978-3-662-48350-3_59(705-716)Online publication date: 12-Nov-2015
https://doi.org/10.1007/978-3-662-48350-3_59
Pellerin JL�vy BCaumon GBotella A(2014)Automatic surface remeshing of 3D structural models at specified resolution: A method based on Voronoi diagramsComputers & Geosciences10.1016/j.cageo.2013.09.00862(103-116)Online publication date: Jan-2014
https://doi.org/10.1016/j.cageo.2013.09.008
Tancer M(2012)Intersection Patterns of Convex Sets via Simplicial Complexes: A SurveyThirty Essays on Geometric Graph Theory10.1007/978-1-4614-0110-0_28(521-540)Online publication date: 29-Oct-2012
https://doi.org/10.1007/978-1-4614-0110-0_28

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents