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This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades.
P. Kirchberger, Über Tchebychefsche Annäherungsmethoden , Math. Ann. 57
1903
Earlier work this paper cites.
C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen , Math. Annalen 64
1907
Earlier work this paper cites.
J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , Math. Ann. 83
1921
Earlier work this paper cites.
E. R. Kampen, Komplexe in euklidischen Räumen , Abh. Math. Semin. Univ. Hambg. 9
1933
Earlier work this paper cites.
A. Flores, Über n n -dimensionale Komplexe, die im R 2 n + 1 R_{2n+1} absolut selbstverschlungen sind , Ergebnisse eines Math. Kolloquiums 6
1934
Earlier work this paper cites.
H. Whitney, The self-intersections of a smooth n n -manifold in 2 n 2n -space , Ann. of Math. (2) 45
1944
Earlier work this paper cites.
B. J. Birch, On 3 N 3N points in a plane , Proc. Cambridge Philos. Soc. 55
1959
Earlier work this paper cites.
H. Tverberg, A generalization of Radon’s theorem , J. London Math. Soc. 41
1966
Earlier work this paper cites.
J. R. Reay, An extension of Radon’s theorem , Illinois J. Math. 12
1968
Earlier work this paper cites.
D. G. Larman, On Sets Projectively Equivalent to the Vertices of a Convex Polytope , Bull. Lond. Math. Soc. 4
1972
Earlier work this paper cites.
B. Lindström, A theorem on families of sets , J. Combinatorial Theory Ser. A 13
1972
Earlier work this paper cites.
J. P. Doignon, Convexity in cristallographical lattices , J. Geom. 3
1973
Earlier work this paper cites.
D. E. Bell, A theorem concerning the integer lattice , Studies in Appl. Math. 56
1976
Earlier work this paper cites.
H. E. Scarf, An observation on the structure of production sets with indivisibilities , Proc. Nat. Acad. Sci. U.S.A. 74
1977
Earlier work this paper cites.
L. Lovász, Kneser’s conjecture, chromatic number, and homotopy , J. Combin. Theory, Ser. A 25
1978
Earlier work this paper cites.
E. G. Bajmóczy and I. Bárány, On a common generalization of Borsuk’s and Radon’s theorem , Acta Mathematica Academiae Scientiarum Hungaricae 34
1979
Earlier work this paper cites.
J. Eckhoff, Radon’s theorem revisited , Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978), Birkhäuser, Basel-Boston, Mass., 1979, pp. 164–185
1979
Earlier work this paper cites.
G. Sierksma, Convexity without linearity; the dutch cheese problem , Mimeographed notes, 1979
1979
Earlier work this paper cites.
I. Bárány, S. B. Shlosman, and A. Szűcs, On a topological generalization of a theorem of Tverberg , Journal of the London Mathematical Society 2
1981
Earlier work this paper cites.
R. Jamison, Partition numbers for trees and ordered sets , Pacific J. Math. 96
1981
Earlier work this paper cites.
I. Bárány, A generalization of Carathéodory’s theorem , Discrete Math. 40
1982
Earlier work this paper cites.
A. Dold, Simple proofs of some Borsuk-Ulam results , Contemp. Math 19
1983
Earlier work this paper cites.
P. Erdős and M. Simonovits, Supersaturated graphs and hypergraphs , Combinatorica 3
1983
Earlier work this paper cites.
E. Boros and Z. Füredi, The number of triangles covering the center of an n-set , Geom Dedicata 17
1984
Earlier work this paper cites.
N. Alon, P. Frankl, and L. Lovász, The chromatic number of Kneser hypergraphs , Trans. Amer. Math. Soc. 298
1986
Earlier work this paper cites.
M. Özaydin, Equivariant maps for the symmetric group , unpublished preprint, University of Winsconsin-Madison, 17 pages, 1987
1987
Earlier work this paper cites.
V. L. Dol’nikov, A certain combinatorial inequality , Siberian Math. J. 29
1988
Earlier work this paper cites.
L. Lovász and A. Schrijver, Orthogonal representations and connectivity of graphs , Linear algebra and its applications 114
1989
Earlier work this paper cites.
I. Bárány, Z. Füredi, and L. Lovász, On the number of halving planes , Combinatorica 10
1990
Earlier work this paper cites.
K. S. Sarkaria, A generalized Kneser conjecture , J. Comb. Theory B 49
1990
Earlier work this paper cites.
R. T. Živaljević and S. T. Vrećica, An extension of the ham sandwich theorem , Bull. London Math. Soc. 22
1990
Earlier work this paper cites.
S. Onn, On the geometry and computational complexity of Radon partitions in the integer lattice , SIAM J. Discrete Math. 4
1991
Earlier work this paper cites.
N. Alon, I. Bárány, Z. Füredi, and D. J. Kleitman, Point selections and weak ε \varepsilon -nets for convex hulls , Combin. Probab. Comput. 1
1992
Earlier work this paper cites.
N. Alon and D. J. Kleitman, Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem , Adv. Math. 96
1992
Cited alongside, same era.
I. Bárány and D. G. Larman, A Colored Version of Tverberg’s Theorem , J. London Math. Soc. 45
1992
Cited alongside, same era.
I. Kříž, Equivariant cohomology and lower bounds for chromatic numbers , Trans. Amer. Math. Soc. 333
1992
Cited alongside, same era.
R. T. Živaljević and S. T. Vrećica, The colored Tverberg’s problem and complexes of injective functions , J. Combin. Theory, Ser. A 61
1992
Cited alongside, same era.
D. Avis, The m m -core properly contains the m m -divisible pints in space , Pattern recognition Letters 14
1993
Cited alongside, same era.
M. Gromov, Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , Geom. Func. Anal. 20
2010
Later among the works it cites.
G. L. Miller and D. R. Sheehy, Approximate centerpoints with proofs , Comput. Geom. 43
2010
Later among the works it cites.
J. L. Arocha, J. Bracho, L. Montejano, and J. L. Ramírez-Alfonsín, Transversals to the convex hulls of all k k -sets of discrete subsets of R n R^{n} , J. Combin. Theory, Ser. A 118
2011
Later among the works it cites.
P. V. M. Blagojević, B. Matschke, and G. M. Ziegler, Optimal bounds for a colorful Tverberg-Vrećica type problem , Adv. Math. 226
2011
Later among the works it cites.
P. Soberón and R. Strausz, A generalisation of Tverberg’s theorem , Discrete Comput. Geom. 47
2012
Later among the works it cites.
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1993
Cited alongside, same era.
A. Vučić and R. T. Živaljević, Note on a conjecture of Sierksma , Discrete Comput. Geom. 9
1993
Cited alongside, same era.
S. T. Vrećica and R. T. Živaljević, New cases of the colored Tverberg theorem , Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 325–334
1994
Cited alongside, same era.
A.Y. Volovikov, On a topological generalization of the Tverberg theorem , Math Notes 59
1996
Cited alongside, same era.
I. Bárány and S. Onn, Colourful linear programming and its relatives , Math. Oper. Res. 22
1997
Cited alongside, same era.
A. Pór, Colorful theorems in convexity , 1997, diploma thesis, Eötovös University, Budapest
1997
Cited alongside, same era.
J. Pach, A Tverberg-type result on multicolored simplices , Comput. Geom. 10
1998
Cited alongside, same era.
W. Mulzer and D. Werner, Approximating Tverberg points in linear time for any fixed dimension , Discrete Comput. Geom. 50
2013
Later among the works it cites.
P. V. M. Blagojević, F. Frick, and G. M. Ziegler, Tverberg plus constraints , Bull. Lond. Math. Soc. 46
2014
Later among the works it cites.
W. Mulzer and Y. Stein, Algorithms for tolerant Tverberg partitions , Internat. J. Comput. Geom. Appl. 24
2014
Later among the works it cites.
2015
Later among the works it cites.
F. Frick, Counterexamples to the topological Tverberg conjecture , Oberwolfach Reports 12
2015
Later among the works it cites.
2015
Later among the works it cites.
P. Soberón, Equal coefficients and tolerance in coloured Tverberg partitions , Combinatorica 35
2015
Later among the works it cites.
2016
Later among the works it cites.
I. Bárány, P. V. M. Blagojević, and G. M. Ziegler, Tverberg’s theorem at 50: extensions and counterexamples , Notices Amer. Math. Soc. 63
2016
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2016
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2016
Later among the works it cites.
A. F. Holmsen, The intersection of a matroid and an oriented matroid , Adv. Math. 290
2016
Later among the works it cites.
2016
Later among the works it cites.
M. A. Perles and M. Sigron, Some variations on Tverberg’s theorem , Israel J. Math. 216
2016
Later among the works it cites.
2016
Later among the works it cites.
S. Simon, Average-value Tverberg partitions via finite Fourier analysis , Israel J. Math. 216
2016
Later among the works it cites.
N. Amenta, J. A. De Loera, and P. Soberón, Helly’s theorem: new variations and applications , Algebraic and geometric methods in discrete mathematics, Contemp. Math., vol. 685, Amer. Math. Soc., Providence, RI, 2017, pp. 55–95
2017
Closest in time.
2017
Closest in time.
I. Bárány, G. Kalai, and R. Meshulam, A Tverberg type theorem for matroids , Journey Through Discrete Mathematics. A Tribute to Jiří Matoušek (Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, eds.), Springer, 2017, pp. 115–121
2017
Closest in time.
B. Bukh and G. Nivasch, One-sided epsilon-approximants , A Journey Through Discrete Mathematics, Springer, 2017, pp. 343–356
2017
Closest in time.
I. Bárány and P. Soberón, Tverberg plus minus , arXiv preprint arXiv:1612.05630 (2017)
2017
Closest in time.
P. V. M. Blagojević and G. M. Ziegler, Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story , Journey Through Discrete Mathematics. A Tribute to Jiří Matoušek (Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, eds.), Springer, 2017, pp. 273–341
2017
Closest in time.
J. Chappelon, L. Martínez-Sandoval, L. Montejano, L. P. Montejano, and J. L. Ramírez-Alfonsín, Complete Kneser transversals , Adv. in Appl. Math. 82
2017
Closest in time.
2017
Closest in time.
J. A. De Loera, R. N. La Haye, D. Rolnick, and P. Soberón, Quantitative combinatorial geometry for continuous parameters , Discrete Comput. Geom. 57
2017
Closest in time.
N. García-Colín, M. Raggi, and E. Roldán-Pensado, A note on the tolerant Tverberg theorem , Discrete Comput. Geom. 58
2017
Closest in time.
A. Magazinov and P. Soberón, Positive-fraction intersection results and variations of weak epsilon-nets , Monatsh. Math. 183
2017
Closest in time.
M. J. White, On Tverberg partitions , Israel J. Math. 219
2017
Closest in time.