Tverberg’s theorem is 50 years old: A survey

This survey presents an overview of the advances around Tverberg’s theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg’s theorem and its applications. The survey contains several open problems and conjectures.

[1]  Helge Tverberg A combinatorial mathematician in Norway: some personal reflections , 2001, Discret. Math..

[2]  Noga Alon,et al.  Point Selections and Weak ε-Nets for Convex Hulls , 1992, Combinatorics, Probability and Computing.

[3]  V. Dol'nikov,et al.  A certain combinatorial inequality , 1988 .

[4]  János Pach A Tverberg-type result on multicolored simplices , 1998, Comput. Geom..

[5]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory A.

[6]  Timothy M. Chan An optimal randomized algorithm for maximum Tukey depth , 2004, SODA '04.

[7]  Pablo Soberón Equal coefficients and tolerance in coloured tverberg partitions , 2013, SoCG '13.

[8]  Jesús A. De Loera,et al.  Quantitative Combinatorial Geometry for Continuous Parameters , 2016, Discret. Comput. Geom..

[9]  Attila Pór,et al.  An Improvement on the Rado Bound for the Centerline Depth , 2016, Discret. Comput. Geom..

[10]  J. Radon Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[11]  Herbert E. Scarf An observation on the structure of production sets with indivisibilities , 1977 .

[12]  Florian Frick,et al.  Intersection patterns of finite sets and of convex sets , 2016, 1607.01003.

[13]  Imre Bárány,et al.  A generalization of carathéodory's theorem , 1982, Discret. Math..

[14]  Leonardo Martínez-Sandoval,et al.  Complete Kneser Transversals , 2015, Adv. Appl. Math..

[15]  Jean-Pierre Roudneff,et al.  Partitions of Points into Simplices withk-dimensional Intersection. Part I: The Conic Tverberg's Theorem , 2001, Eur. J. Comb..

[16]  Wolfgang Mulzer,et al.  Algorithms for Tolerant Tverberg Partitions , 2014, Int. J. Comput. Geom. Appl..

[17]  B. J. Birch,et al.  On 3N points in a plane , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  R. Živaljević,et al.  The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles , 1999 .

[19]  Imre Bárány,et al.  On a Topological Generalization of a Theorem of Tverberg , 1981 .

[20]  Ulrich Wagner,et al.  On k-sets and applications , 2003 .

[21]  Miklós Simonovits,et al.  Supersaturated graphs and hypergraphs , 1983, Comb..

[22]  Shmuel Onn,et al.  On the Geometry and Computational Complexity of Radon Partitions in the Integer Lattice , 1991, SIAM J. Discret. Math..

[23]  David E. Bell A Theorem Concerning the Integer Lattice , 1977 .

[24]  An elementary proof of Tverberg’s theorem , 2009 .

[25]  Jürgen Eckhoff,et al.  Radon’s theorem revisited , 1979 .

[26]  Jean-Paul Doignon,et al.  Convexity in cristallographical lattices , 1973 .

[27]  Gary L. Miller,et al.  Approximate centerpoints with proofs , 2010, Comput. Geom..

[28]  Jean-Pierre Roudneff,et al.  Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay's Conjecture in Dimensions 4 and 5 , 2001, Eur. J. Comb..

[29]  Rade T. Zivaljevic,et al.  Note on a conjecture of sierksma , 1993, Discret. Comput. Geom..

[30]  I. Bárány,et al.  A Colored Version of Tverberg's Theorem , 1992 .

[31]  Stephan Hell On the Number of Colored Birch and Tverberg Partitions , 2014, Electron. J. Comb..

[32]  Roman N. Karasev Tverberg's Transversal Conjecture and Analogues of Nonembeddability Theorems for Transversals , 2007, Discret. Comput. Geom..

[33]  Karanbir S. Sarkaria,et al.  A generalized kneser conjecture , 1990, J. Comb. Theory, Ser. B.

[34]  Florian Frick,et al.  On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture , 2016, Electron. J. Comb..

[35]  E. R. Kampen Komplexe in euklidischen Räumen , 1933 .

[36]  Miguel Raggi,et al.  A Note on the Tolerant Tverberg Theorem , 2017, Discret. Comput. Geom..

[37]  Z. Füredi,et al.  The number of triangles covering the center of an n-set , 1984 .

[38]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[39]  Ruy Fabila Monroy,et al.  Very Colorful Theorems , 2009, Discret. Comput. Geom..

[40]  V. Dol'nikov,et al.  A generalization of the ham sandwich theorem , 1992 .

[41]  Stephan Hell,et al.  Tverberg's theorem with constraints , 2007, J. Comb. Theory, Ser. A.

[42]  Imre Bárány,et al.  Colourful Linear Programming and its Relatives , 1997, Math. Oper. Res..

[43]  Florian Frick,et al.  Tverberg plus constraints , 2014, 1401.0690.

[44]  Mia Hubert,et al.  Depth in an Arrangement of Hyperplanes , 1999, Discret. Comput. Geom..

[45]  P. Soberón,et al.  Positive-fraction intersection results and variations of weak epsilon-nets , 2015, 1506.02191.

[46]  Pavle V. M. Blagojevic,et al.  Tverberg-Type Theorems for Matroids: A Counterexample and a Proof , 2017, Combinatorica.

[47]  Jürgen Eckhoff,et al.  The partition conjecture , 2000, Discrete Mathematics.

[48]  Florian Frick,et al.  Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints , 2015, Journal of the European Mathematical Society.

[49]  Stéphane Gaubert,et al.  Carathéodory, Helly and the Others in the Max-Plus World , 2010, Discret. Comput. Geom..

[50]  Bernt Lindström A Theorem on Families of Sets , 1972, J. Comb. Theory, Ser. A.

[51]  Roman N. Karasev,et al.  Tverberg-Type Theorems for Intersecting by Rays , 2010, Discret. Comput. Geom..

[52]  Jesús A. De Loera,et al.  Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets , 2016, Discret. Comput. Geom..

[53]  Robert E. Jamison-Waldner PARTITION NUMBERS FOR TREES AND ORDERED SETS , 1981 .

[54]  Imre Bárány,et al.  Tverberg Plus Minus , 2018, Discret. Comput. Geom..

[55]  Jiří Matoušek,et al.  On the chromatic number of Kneser hypergraphs , 2002 .

[56]  Florian Frick,et al.  Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems , 2017, International Mathematics Research Notices.

[57]  A. Volovikov,et al.  On a topological generalization of the Tverberg theorem , 1996 .

[58]  E. D. Giorgi Selected Papers , 2006 .

[59]  Jorge L. Ramírez Alfonsín Lawrence Oriented Matroids and a Problem of McMullen on Projective Equivalences of Polytopes , 2001, Eur. J. Comb..

[60]  Attila Por Universality of vector sequences and universality of Tverberg partitions , 2018 .

[61]  Pavle V. M. Blagojevi'c,et al.  Optimal bounds for the colored Tverberg problem , 2009, 0910.4987.

[62]  G. Kalai,et al.  A Tverberg type theorem for matroids , 2016, 1607.01599.

[63]  Роман Николаевич Карасeв,et al.  Двойственные теоремы о центральной точке и их обобщения@@@Dual theorems on central points and their generalizations , 2008 .

[64]  Roman Karasev,et al.  Dual theorems on central points and their generalizations , 2008 .

[65]  Murad Ozaydin,et al.  Equivariant Maps for the Symmetric Group , 1987 .

[66]  Jirí Matousek,et al.  Stabbing Simplices by Points and Flats , 2008, Discret. Comput. Geom..

[67]  Sinisa T. Vrecica Tverberg's Conjecture , 2003, Discret. Comput. Geom..

[68]  Jean-Pierre Roudneff New cases of Reay's conjecture on partitions of points into simplices with k-dimensional intersection , 2009, Eur. J. Comb..

[69]  Aart Blokhuis,et al.  The Radon Number of the Three-Dimensional Integer Lattice , 2003, Discret. Comput. Geom..

[70]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[71]  L. Lovász,et al.  Orthogonal representations and connectivity of graphs , 1989 .

[72]  Imre Bárány,et al.  Tverberg’s Theorem at 50: Extensions and Counterexamples , 2016 .

[73]  Imre Bárány Helge Tverberg is eighty: A personal tribute , 2017, Eur. J. Comb..

[74]  Helge Tverberg,et al.  On Generalizations of Radon's Theorem and the Ham Sandwich Theorem , 1993, Eur. J. Comb..

[75]  K. S. Sarkaria A generalized van Kampen-Flores theorem , 1991 .

[76]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems , 2015, ArXiv.

[77]  Boris Bukh Radon partitions in convexity spaces , 2010, ArXiv.

[78]  David Avis The m-core properly contains the m-divisible points in space , 1993, Pattern Recognit. Lett..

[79]  Benjamin Matschke,et al.  Optimal bounds for a colorful Tverberg--Vrecica type problem , 2009, 0911.2692.

[80]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[81]  Gabriel Nivasch,et al.  Classifying unavoidable Tverberg partitions , 2016, Journal of Computational Geometry.

[82]  Roman N. Karasev,et al.  A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem , 2010, Discret. Comput. Geom..

[83]  Helge Tverberg A generalization of Radon's theorem II , 1981, Bulletin of the Australian Mathematical Society.

[84]  Steven Simon Average-value tverberg partitions via finite fourier analysis , 2015 .

[85]  Pablo Soberón Robust Tverberg and Colourful Carathéodory Results via Random Choice , 2018, Comb. Probab. Comput..

[86]  On Tverberg partitions , 2015, 1508.07262.

[87]  J. Eckhoff Helly, Radon, and Carathéodory Type Theorems , 1993 .

[88]  Pavle V. M. Blagojevi'c,et al.  Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story , 2016, 1605.07321.

[89]  Heather A. Harrington,et al.  Algebraic and Geometric Methods in Discrete Mathematics , 2017 .

[90]  Jirí Matousek,et al.  Lower bounds for weak epsilon-nets and stair-convexity , 2008, SCG '09.

[91]  A. Dold,et al.  Simple proofs of some Borsuk - Ulam results , 1983 .

[92]  H. Whitney The Self-Intersections of a Smooth n-Manifold in 2n-Space , 1944 .

[93]  I. Bárány,et al.  On a common generalization of Borsuk's and Radon's theorem , 1979 .

[94]  Gabriel Nivasch,et al.  One-sided epsilon-approximants , 2016, ArXiv.

[95]  Xavier Goaoc,et al.  The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg , 2017, Bulletin of the American Mathematical Society.

[96]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, III. Codimension 2 , 2015, Israel Journal of Mathematics.

[97]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[98]  Stephan Hell On the number of Tverberg partitions in the prime power case , 2007, Eur. J. Comb..

[99]  Stephan Hell On the Number of Birch Partitions , 2008, Discret. Comput. Geom..

[100]  C. Carathéodory Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen , 1907 .

[101]  Pablo Soberón Equal coefficients and tolerance in coloured Tverberg partitions , 2015, Comb..

[102]  Paul Kirchberger,et al.  Über Tchebychefsche Annäherungsmethoden , 1903 .

[103]  J. R. Reay Several generalizations of Tverberg’s theorem , 1979 .

[104]  Gil Kalai,et al.  A topological colorful Helly theorem , 2005 .

[105]  David Forge,et al.  10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope , 2001, Eur. J. Comb..

[106]  R. Živaljević,et al.  An Extension of the Ham Sandwich Theorem , 1990 .

[107]  G. Kalai Combinatorics with a Geometric Flavor , 2000 .

[108]  N. Alon,et al.  Piercing convex sets and the hadwiger-debrunner (p , 1992 .

[109]  Florian Frick,et al.  Counterexamples to the topological Tverberg conjecture , 2015 .

[110]  Micha A. Perles,et al.  Some variations on Tverberg’s theorem , 2016 .

[111]  H. Tverberg A Generalization of Radon's Theorem , 1966 .

[112]  K. S. Sarkaria Tverberg’s theorem via number fields , 1992 .

[113]  Zoltán Füredi,et al.  On the number of halving planes , 1989, SCG '89.

[114]  I. Kríz A correction to “Equivariant cohomology and lower bounds for chromatic numbers” , 1992 .

[115]  P. Soberón Helly‐type theorems for the diameter , 2015, 1509.07908.

[116]  D. G. Larman On Sets Projectively Equivalent to the Vertices of a Convex Polytope , 1972 .

[117]  P. Soberón Tverberg partitions as epsilon-nets , 2017 .

[118]  David Rolnick,et al.  Quantitative (p, q) theorems in combinatorial geometry , 2017, Discret. Math..

[119]  John R. Reay An extension of Radon's theorem , 1968 .

[120]  Andreas F. Holmsen The intersection of a matroid and an oriented matroid , 2016 .

[121]  M. Gromov Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[122]  Pablo Soberón,et al.  A Generalisation of Tverberg’s Theorem , 2012, Discret. Comput. Geom..

[123]  Pavle V. M. Blagojevi'c,et al.  The Topological Transversal Tverberg Theorem Plus Constraints , 2016, 1604.02814.

[124]  Jorge L. Ramírez Alfonsín,et al.  TRANSVERSALS TO THE CONVEX HULLS OF ALL k -SETS OF DISCRETE SUBSETS OF R n , 2013 .