## Tverberg’s theorem is 50 years old: a survey.(English)Zbl 1401.52012

The theorem of H. Tverberg [J. Lond. Math. Soc. 41, 123–128 (1966; Zbl 0131.20002)] says that any set of $$(r-1)(d+1)+1$$ points in $${\mathbb R}^d$$ can be partitioned into $$r$$ parts whose convex hulls intersect. This result has had, and still has, considerable impact on combinatorial convexity. The present paper gives a survey, with emphasis on the developments of the last two decades. It starts with two modern proofs of Tverberg’s theorem, due respectively to J.-P. Roudneff [Eur. J. Comb. 22, No. 5, 745–765 (2001; Zbl 1007.52006)] and K. S. Sarkaria [Isr. J. Math. 79, No. 2–3, 317–320 (1992; Zbl 0786.52005)]. Section 2 deals with topological versions and relations to the generalized Van Kampen-Flores Theorem. The Topological Tverberg Theorem says: If $$f$$ is a continuous map from the $$d$$-skeleton of the $$n$$-simplex $$\Delta^n$$ to $${\mathbb R}^d$$, with $$n=(r-1)(d+1)$$ and $$r$$ a prime power, then there are disjoint faces $$F_1,\dots,F_r$$ of $$\Delta^n$$ such that $$\bigcap_{j=1}^r f(F_j)\not=\emptyset$$. After it was an open problem for decades, it was proved recently that this theorem does not hold if $$r$$ is not a prime power. Section 3 treats various colorful versions and extensions of Tverberg’s theorem, and also mentions some open problems. Section 4 deals with several questions on the structure of Tverberg partitions, such as their minimal number, resistance to changes, modifications of the intersection condition, conjectures by Sierksma, Tverberg-Vrećica, Reay, and some new conjectures and open problems. Section 5 continues with variations and conjectures around Tverberg’s theorem and treats, in particular, universal Tverberg partitions. Various applications of Tverberg’s theorem to combinatorial geometry are the topic of Section 6, for example, the weak $$\varepsilon$$-net theorem for convex sets and the relation of Tverberg type results to Kneser hypergraphs. Section 7 is concerned with variations of Tverberg’s theorem if the underlying setting is changed; examples are integer coordinates, convexity spaces, quantitative versions. Altogether, this is a comprehensive and attractive survey on a great theorem and its aftermath.

### MSC:

 52A35 Helly-type theorems and geometric transversal theory 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry 52-03 History of convex and discrete geometry 01A60 History of mathematics in the 20th century 52A37 Other problems of combinatorial convexity

### Keywords:

Tverberg’s theorem; combinatorial convexity

### Citations:

Zbl 0131.20002; Zbl 1007.52006; Zbl 0786.52005
Full Text:

### References:

 [1] Alon, N.; B{\'a}r{\'a}ny, I.; F{\"u}redi, Z.; Kleitman, D. J., Point selections and weak $$\varepsilon$$-nets for convex hulls, Combin. Probab. Comput., 1, 03, 189-200 (1992) · Zbl 0797.52004 [2] Alon, N.; Frankl, P.; Lov\'asz, L., The chromatic number of Kneser hypergraphs, Trans. Amer. Math. Soc., 298, 1, 359-370 (1986) · Zbl 0605.05033 [3] Alon, Noga; Kleitman, Daniel J., Piercing convex sets and the Hadwiger-Debrunner $$(p,q)$$-problem, Adv. Math., 96, 1, 103-112 (1992) · Zbl 0768.52001 [4] Amenta, Nina; De Loera, Jes\'us A.; Sober\'on, Pablo, Helly’s theorem: new variations and applications. Algebraic and geometric methods in discrete mathematics, Contemp. Math. 685, 55-95 (2017), Amer. Math. Soc., Providence, RI · Zbl 1383.52006 [5] Arocha, Jorge L.; B\'ar\'any, Imre; Bracho, Javier; Fabila, Ruy; Montejano, Luis, Very colorful theorems, Discrete Comput. Geom., 42, 2, 142-154 (2009) · Zbl 1173.52002 [6] Arocha, J. L.; Bracho, J.; Montejano, L.; Ram\'\i rez Alfons\'\i n, J. L., Transversals to the convex hulls of all $$k$$-sets of discrete subsets of $$\mathbb{R}^n$$, J. Combin. Theory Ser. A, 118, 1, 197-207 (2011) · Zbl 1231.05046 [7] Asada, M.; Chen, R.; Frick, F.; Huang, F.; Polevy, M.; Stoner, D.; Tsang, L. H.; Wellner, Z., On Reay’s relaxed Tverberg conjecture and generalizations of Conway’s thrackle conjecture, arXiv:1608.04279 (2016) · Zbl 1397.52004 [8] Avis, D., The $$m$$-core properly contains the $$m$$-divisible pints in space, Pattern Recognition Letters, 14, 703-705 (1993) · Zbl 0781.68110 [9] Avvakumov, S.; Mabillard, I.; Skopenkov, A.; Wagner, U., Eliminating higher-multiplicity intersections, III. Codimension $$2$$, arXiv:1511.03501 (2015) · Zbl 1486.57035 [10] Bajm\'oczy, E. G.; B\'ar\'any, I., On a common generalization of Borsuk’s and Radon’s theorem, Acta Math. Acad. Sci. Hungar., 34, 3-4, 347-350 (1980) (1979) · Zbl 0446.52010 [11] B\'ar\'any, Imre, A generalization of Carath\'eodory’s theorem, Discrete Math., 40, 2-3, 141-152 (1982) · Zbl 0492.52005 [12] B\'ar\'any, Imre, Helge Tverberg is eighty: a personal tribute, European J. Combin., 66, 24-27 (2017) · Zbl 1372.01024 [13] B\'ar\'any, Imre; Blagojevi\'c, Pavle V. M.; Ziegler, G\“unter M., Tverberg”s theorem at 50: extensions and counterexamples, Notices Amer. Math. Soc., 63, 7, 732-739 (2016) · Zbl 1358.52022 [14] B{\'a}r{\'a}ny, I.; F{\“u}redi, Z.; Lov{\'”a}sz, L., On the number of halving planes, Combinatorica, 10, 175-183 (1990) · Zbl 0718.52009 [15] B\'ar\'any, Imre; Kalai, Gil; Meshulam, Roy, A Tverberg type theorem for matroids. A journey through discrete mathematics, 115-121 (2017), Springer, Cham · Zbl 1387.05036 [16] B\'ar\'any, I.; Larman, D. G., A colored version of Tverberg’s theorem, J. London Math. Soc. (2), 45, 2, 314-320 (1992) · Zbl 0769.52008 [17] B\'ar\'any, Imre; Onn, Shmuel, Colourful linear programming and its relatives, Math. Oper. Res., 22, 3, 550-567 (1997) · Zbl 0887.90111 [18] B\'ar\'any, I.; Shlosman, S. B.; Sz\H ucs, A., On a topological generalization of a theorem of Tverberg, J. London Math. Soc. (2), 23, 1, 158-164 (1981) · Zbl 0453.55003 [19] B\'ar\'any, I.; Sober\'on, P., Tverberg plus minus, arXiv:1612.05630 (2017) · Zbl 1401.52014 [20] Bell, David E., A theorem concerning the integer lattice, Studies in Appl. Math., 56, 2, 187-188 (1976/77) · Zbl 0388.90051 [21] Bezdek, K\'aroly; Blokhuis, Aart, The Radon number of the three-dimensional integer lattice, Discrete Comput. Geom., 30, 2, 181-184 (2003) · Zbl 1043.52010 [22] Birch, B. J., On $$3N$$ points in a plane, Proc. Cambridge Philos. Soc., 55, 289-293 (1959) · Zbl 0089.38502 [23] Blagojevi\'c, P. V. M.; Dimitrijevi\'c Blagojevi\'cA. S.; Ziegler, G. M., The topological transversal {Tverberg} theorem plus constraints, http://arxiv.org/abs/1604.02814arXiv:1604.02814 (2016) [24] Blagojevi\'c, Pavle V. M.; Frick, Florian; Ziegler, G\"unter M., Tverberg plus constraints, Bull. Lond. Math. Soc., 46, 5, 953-967 (2014) · Zbl 1305.52021 [25] Blagojevi{\'c}, P. V. M.; Frick, F.; Ziegler, G. M., Barycenters of Polytope Skeleta and Counterexamples to the Topological Tverberg Conjecture, via Constraints, http://arxiv.org/abs/1510.07984 (2017) · Zbl 1428.52008 [26] Blagojevi{\'c}, P. V. M.; Haase, A.; Ziegler, G. M., Tverberg-type theorems for matroids: A counterexample and a proof, arXiv:1705.03624 (2017) [27] Blagojevi\'c, Pavle V. M.; Matschke, Benjamin; Ziegler, G\“unter M., Optimal bounds for a colorful Tverberg-Vre\'”cica type problem, Adv. Math., 226, 6, 5198-5215 (2011) · Zbl 1213.52009 [28] Blagojevi\'c, Pavle V. M.; Matschke, Benjamin; Ziegler, G\"unter M., Optimal bounds for the colored Tverberg problem, J. Eur. Math. Soc. (JEMS), 17, 4, 739-754 (2015) · Zbl 1327.52009 [29] Blagojevi\'c, P. V. M.; Ziegler, G. M., Beyond the Borsuk-Ulam Theorem: The Topological Tverberg Story, Journey through discrete mathematics. a tribute to ji\v{r}\'{\i} matou\v{s}ek, 273-341 (2017), Springer · Zbl 1470.52007 [30] Boros, E.; F\"uredi, Z., The number of triangles covering the center of an $$n$$-set, Geom. Dedicata, 17, 1, 69-77 (1984) · Zbl 0595.52002 [31] Bukh, B., Radon partitions in convexity spaces, arXiv:1009.2384 (2010) [32] Bukh, Boris; Loh, Po-Shen; Nivasch, Gabriel, Classifying unavoidable Tverberg partitions, J. Comput. Geom., 8, 1, 174-205 (2017) · Zbl 1397.52011 [33] Bukh, Boris; Matou\v sek, Ji\v r\'\i; Nivasch, Gabriel, Stabbing simplices by points and flats, Discrete Comput. Geom., 43, 2, 321-338 (2010) · Zbl 1186.52001 [34] Bukh, Boris; Matou\v sek, Ji\v r\'\i; Nivasch, Gabriel, Lower bounds for weak epsilon-nets and stair-convexity, Israel J. Math., 182, 199-208 (2011) · Zbl 1222.68395 [35] Bukh, Boris; Nivasch, Gabriel, One-sided epsilon-approximants. A journey through discrete mathematics, 343-356 (2017), Springer, Cham · Zbl 1387.05178 [36] Carath\'eodory, C., \“Uber den Variabilit\'”atsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann., 64, 1, 95-115 (1907) · JFM 38.0448.01 [37] Chan, Timothy M., An optimal randomized algorithm for maximum Tukey depth. Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 430-436 (2004), ACM, New York · Zbl 1317.68246 [38] Chappelon, J.; Mart\'\i nez-Sandoval, L.; Montejano, L.; Montejano, L. P.; Ram\'\i rez Alfons\'\i n, J. L., Complete Kneser transversals, Adv. in Appl. Math., 82, 83-101 (2017) · Zbl 1352.52017 [39] De Loera, J. A.; Goaoc, X.; Meunier, F.; Mustafa, N., The discrete yet ubiquitous theorems of Carath\'eodory, Helly, Sperner, Tucker, and Tverberg, arXiv:1706.05975 (2017) · Zbl 1460.52010 [40] De Giorgi, Ennio, Selected papers, Springer Collected Works in Mathematics, x+888 pp. (2013), Springer, Heidelberg · Zbl 1279.00037 [41] De Loera, Jes\'us A.; La Haye, Reuben N.; Rolnick, David; Sober\'on, Pablo, Quantitative combinatorial geometry for continuous parameters, Discrete Comput. Geom., 57, 2, 318-334 (2017) · Zbl 1369.52012 [42] De Loera, Jesus A.; La Haye, Reuben N.; Rolnick, David; Sober\'on, Pablo, Quantitative Tverberg theorems over lattices and other discrete sets, Discrete Comput. Geom., 58, 2, 435-448 (2017) · Zbl 1386.52004 [43] Doignon, Jean-Paul, Convexity in cristallographical lattices, J. Geometry, 3, 71-85 (1973) · Zbl 0245.52004 [44] Dold, Albrecht, Simple proofs of some Borsuk-Ulam results. Proceedings of the Northwestern Homotopy Theory Conference, Evanston, Ill., 1982, Contemp. Math. 19, 65-69 (1983), Amer. Math. Soc., Providence, RI · Zbl 0521.55002 [45] Dol’nikov, V. L., A certain combinatorial inequality, Siberian Math. J., 29, 3, 375-379 (1988) · Zbl 0675.05031 [46] Dol\cprime nikov, V. L., A generalization of the sandwich theorem, Mat. Zametki. Math. Notes, 52 52, 1-2, 771-779 (1993) (1992) · Zbl 0787.52003 [47] Eckhoff, J\“urgen, Radon”s theorem revisited. Contributions to geometry, Proc. Geom. Sympos., Siegen, 1978, 164-185 (1979), Birkh\"auser, Basel-Boston, Mass. [48] Eckhoff, J\“urgen, Helly, Radon, and Carath\'”eodory type theorems. Handbook of convex geometry, Vol.A, B, 389-448 (1993), North-Holland, Amsterdam · Zbl 0791.52009 [49] Eckhoff, J\"urgen, The partition conjecture, Discrete Math., 221, 1-3, 61-78 (2000) · Zbl 0971.52008 [50] Erd\H os, Paul; Simonovits, Mikl\'os, Supersaturated graphs and hypergraphs, Combinatorica, 3, 2, 181-192 (1983) · Zbl 0529.05027 [51] Flores, A., \"Uber $$n$$-dimensionale {K}omplexe, die im {$$R_{2n+1}$$} absolut selbstverschlungen sind, Ergebnisse eines Math. Kolloquiums, 6, 4-7 (1933/1934) [52] Forge, David; Las Vergnas, Michel; Schuchert, Peter, 10 points in dimension 4 not projectively equivalent to the vertices of a convex polytope, European J. Combin., 22, 5, 705-708 (2001) · Zbl 0984.52017 [53] Frick, F., Counterexamples to the topological Tverberg conjecture, Oberwolfach Reports, 12, 1, 318-321 (2015) [54] Frick, F., Chromatic numbers of stable Kneser hypergraphs via topological Tverberg-type theorems, arXiv:1710.09434 (2017) [55] Frick, Florian, Intersection patterns of finite sets and of convex sets, Proc. Amer. Math. Soc., 145, 7, 2827-2842 (2017) · Zbl 1360.05058 [56] Garc\'\i a-Col\'\i n, Natalia; Raggi, Miguel; Rold\'an-Pensado, Edgardo, A note on the tolerant Tverberg theorem, Discrete Comput. Geom., 58, 3, 746-754 (2017) · Zbl 1377.52010 [57] Gaubert, St\'ephane; Meunier, Fr\'ed\'eric, Carath\'eodory, Helly and the others in the max-plus world, Discrete Comput. Geom., 43, 3, 648-662 (2010) · Zbl 1219.14071 [58] Gromov, Mikhail, Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal., 20, 2, 416-526 (2010) · Zbl 1251.05039 [59] Gr\“unbaum, Branko, Convex polytopes, Graduate Texts in Mathematics 221, xvi+468 pp. (2003), Springer-Verlag, New York · Zbl 1024.52001 [60] Hell, Stephan, On the number of Tverberg partitions in the prime power case, European J. Combin., 28, 1, 347-355 (2007) · Zbl 1121.05007 [61] Hell, Stephan, On the number of Birch partitions, Discrete Comput. Geom., 40, 4, 586-594 (2008) · Zbl 1169.05008 [62] Hell, Stephan, Tverberg’s theorem with constraints, J. Combin. Theory Ser. A, 115, 8, 1402-1416 (2008) · Zbl 1221.05024 [63] Hell, Stephan, On the number of colored Birch and Tverberg partitions, Electron. J. Combin., 21, 3, Paper 3.23, 12 pp. (2014) · Zbl 1300.05037 [64] Holmsen, Andreas F., The intersection of a matroid and an oriented matroid, Adv. Math., 290, 1-14 (2016) · Zbl 1329.05058 [65] Jamison-Waldner, Robert E., Partition numbers for trees and ordered sets, Pacific J. Math., 96, 1, 115-140 (1981) · Zbl 0482.52010 [66] Kalai, Gil, Combinatorics with a geometric flavor, Geom. Funct. Anal., Special Volume, 742-791 (2000) · Zbl 0989.05001 [67] Kalai, Gil; Meshulam, Roy, A topological colorful Helly theorem, Adv. Math., 191, 2, 305-311 (2005) · Zbl 1064.52008 [68] van Kampen, E. R., Komplexe in euklidischen R\"aumen, Abh. Math. Sem. Univ. Hamburg, 9, 1, 72-78 (1933) · JFM 58.0615.02 [69] Karasev, Roman N., Tverberg’s transversal conjecture and analogues of nonembeddability theorems for transversals, Discrete Comput. Geom., 38, 3, 513-525 (2007) · Zbl 1163.52301 [70] Karas\"ev, R. N., Dual theorems on a central point and their generalizations, Mat. Sb.. Sb. Math., 199 199, 9-10, 1459-1479 (2008) · Zbl 1159.52008 [71] Karasev, R. N., Tverberg-type theorems for intersecting by rays, Discrete Comput. Geom., 45, 2, 340-347 (2011) · Zbl 1213.52005 [72] Karasev, Roman, A simpler proof of the Boros-F\“uredi-B\'”ar\'any-Pach-Gromov theorem, Discrete Comput. Geom., 47, 3, 492-495 (2012) · Zbl 1237.05054 [73] Kirchberger, Paul, \`“Uber Tchebychefsche Ann\'”aherungsmethoden, Math. Ann., 57, 4, 509-540 (1903) · JFM 34.0438.01 [74] Knill, Emanuel; Laflamme, Raymond; Viola, Lorenza, Theory of quantum error correction for general noise, Phys. Rev. Lett., 84, 11, 2525-2528 (2000) · Zbl 0956.81008 [75] K\v r\'\i\v z, Igor, Equivariant cohomology and lower bounds for chromatic numbers, Trans. Amer. Math. Soc., 333, 2, 567-577 (1992) · Zbl 0756.05055 [76] Larman, D. G., On sets projectively equivalent to the vertices of a convex polytope, Bull. London Math. Soc., 4, 6-12 (1972) · Zbl 0248.52010 [77] Lindstr\"om, Bernt, A theorem on families of sets, J. Combinatorial Theory Ser. A, 13, 274-277 (1972) · Zbl 0243.05005 [78] Lov\'asz, L., Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A, 25, 3, 319-324 (1978) · Zbl 0418.05028 [79] Lov\'asz, L.; Saks, M.; Schrijver, A., Orthogonal representations and connectivity of graphs, Linear Algebra Appl., 114/115, 439-454 (1989) · Zbl 0681.05048 [80] Mabillard, I.; Wagner, U., Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems, arXiv:1508.02349 (2015) [81] Magazinov, A.; P\'or, A., An improvement on the Rado bound for the centerline depth, arXiv:1603.01641 (2016) · Zbl 1385.52019 [82] Magazinov, Alexander; Sober\'on, Pablo, Positive-fraction intersection results and variations of weak epsilon-nets, Monatsh. Math., 183, 1, 165-176 (2017) · Zbl 1456.52009 [83] Matou\v sek, Ji\v r\'\i, Lectures on discrete geometry, Graduate Texts in Mathematics 212, xvi+481 pp. (2002), Springer-Verlag, New York · Zbl 0999.52006 [84] Miller, Gary L.; Sheehy, Donald R., Approximate centerpoints with proofs, Comput. Geom., 43, 8, 647-654 (2010) · Zbl 1206.65101 [85] Mulzer, Wolfgang; Stein, Yannik, Algorithms for tolerant Tverberg partitions, Internat. J. Comput. Geom. Appl., 24, 4, 261-273 (2014) · Zbl 1336.52006 [86] Mulzer, Wolfgang; Werner, Daniel, Approximating Tverberg points in linear time for any fixed dimension, Discrete Comput. Geom., 50, 2, 520-535 (2013) · Zbl 1298.68281 [87] Onn, Shmuel, On the geometry and computational complexity of Radon partitions in the integer lattice, SIAM J. Discrete Math., 4, 3, 436-446 (1991) · Zbl 0735.52007 [88] \"Ozaydin, M., Equivariant maps for the symmetric group (1987) [89] Pach, J\'anos, A Tverberg-type result on multicolored simplices, Comput. Geom., 10, 2, 71-76 (1998) · Zbl 0896.68143 [90] Perles, M. A.; Sigron, M., Some variations on {T}verberg’s theorem, Israel J. Math., 216, 2, 957-972 (2016) · Zbl 1357.52003 [91] P{\'o}r, A., Colorful theorems in convexity (1997) [92] P{\'o}r, A., Universality of vector sequences and universality of Tverberg partitions, arXiv:1805.07197 (2018) [93] Radon, Johann, Mengen konvexer K\"orper, die einen gemeinsamen Punkt enthalten, Math. Ann., 83, 1-2, 113-115 (1921) · JFM 48.0834.04 [94] Ram\'\i rez Alfons\'\i n, J. L., Lawrence oriented matroids and a problem of McMullen on projective equivalences of polytopes, European J. Combin., 22, 5, 723-731 (2001) · Zbl 0984.52019 [95] Reay, John R., An extension of Radon’s theorem, Illinois J. Math., 12, 184-189 (1968) · Zbl 0153.52001 [96] Reay, John R., Several generalizations of Tverberg’s theorem, Israel J. Math., 34, 3, 238-244 (1980) (1979) · Zbl 0428.52002 [97] Rolnick, D.; Sober\'on, P., Algorithms for Tverberg’s theorem via centerpoint theorems, arXiv:1601.03083v2 (2016) [98] Rolnick, David; Sober\'on, Pablo, Quantitative $$(p,q)$$ theorems in combinatorial geometry, Discrete Math., 340, 10, 2516-2527 (2017) · Zbl 1379.52007 [99] Roudneff, Jean-Pierre, Partitions of points into simplices with $$k$$-dimensional intersection. II. Proof of Reay’s conjecture in dimensions 4 and 5, European J. Combin., 22, 5, 745-765 (2001) · Zbl 1007.52006 [100] Roudneff, Jean-Pierre, Partitions of points into simplices with $$k$$-dimensional intersection. II. Proof of Reay’s conjecture in dimensions 4 and 5, European J. Combin., 22, 5, 745-765 (2001) · Zbl 1007.52006 [101] Roudneff, Jean-Pierre, New cases of Reay’s conjecture on partitions of points into simplices with $$k$$-dimensional intersection, European J. Combin., 30, 8, 1919-1943 (2009) · Zbl 1227.05064 [102] Rousseeuw, P. J.; Hubert, M., Depth in an arrangement of hyperplanes, Discrete Comput. Geom., 22, 2, 167-176 (1999) · Zbl 0944.52009 [103] Sarkaria, K. S., A generalized Kneser conjecture, J. Combin. Theory Ser. B, 49, 2, 236-240 (1990) · Zbl 0714.05001 [104] Sarkaria, K. S., A generalized van Kampen-Flores theorem, Proc. Amer. Math. Soc., 111, 2, 559-565 (1991) · Zbl 0722.57007 [105] Sarkaria, K. S., Tverberg’s theorem via number fields, Israel J. Math., 79, 2-3, 317-320 (1992) · Zbl 0786.52005 [106] Scarf, Herbert E., An observation on the structure of production sets with indivisibilities, Proc. Nat. Acad. Sci. U.S.A., 74, 9, 3637-3641 (1977) · Zbl 0381.90081 [107] Sierksma, G., Convexity without linearity; the dutch cheese problem (1979) [108] Simon, Steven, Average-value Tverberg partitions via finite Fourier analysis, Israel J. Math., 216, 2, 891-904 (2016) · Zbl 1361.43002 [109] Sober\'on, Pablo, Equal coefficients and tolerance in coloured Tverberg partitions, Combinatorica, 35, 2, 235-252 (2015) · Zbl 1374.52006 [110] Sober\'on, P., Helly-type theorems for the diameter, Bull. Lond. Math. Soc., 48, 4, 577-588 (2016) · Zbl 1353.52005 [111] Sober\'on, P., Tverberg partitions as epsilon-nets, arXiv:1711.11496 (2017) · Zbl 1438.52017 [112] Sober\'on, Pablo, Robust Tverberg and Colourful Carath\'eodory Results via Random Choice, Combin. Probab. Comput., 27, 3, 427-440 (2018) · Zbl 1387.52012 [113] Sober\'on, Pablo; Strausz, Ricardo, A generalisation of Tverberg’s theorem, Discrete Comput. Geom., 47, 3, 455-460 (2012) · Zbl 1243.52005 [114] Tverberg, H., A generalization of Radon’s theorem, J. London Math. Soc., 41, 123-128 (1966) · Zbl 0131.20002 [115] Tverberg, H., On equal unions of sets, Studies in {P}ure {M}athematics ({P}resented to {R}ichard {R}ado), 249-250 (1971), Academic Press, London · Zbl 0221.05007 [116] Tverberg, H., A generalization of Radon’s theorem. II, Bull. Austral. Math. Soc., 24, 3, 321-325 (1981) · Zbl 0465.52002 [117] Tverberg, Helge, A combinatorial mathematician in Norway: some personal reflections, Discrete Math., 241, 1-3, 11-22 (2001) · Zbl 1016.01026 [118] Tverberg, Helge; Vre\'cica, Sini\v sa, On generalizations of Radon’s theorem and the ham sandwich theorem, European J. Combin., 14, 3, 259-264 (1993) · Zbl 0777.52005 [119] Volovikov, A. Yu., On a topological generalization of Tverberg’s theorem, Mat. Zametki. Math. Notes, 59 59, 3-4, 324-325 (1996) · Zbl 0879.55004 [120] Vre\'cica, Sini\v sa T., Tverberg’s conjecture, Discrete Comput. Geom., 29, 4, 505-510 (2003) · Zbl 1069.52009 [121] Vu\v ci\'c, Aleksandar; \v Zivaljevi\'c, Rade T., Note on a conjecture of Sierksma, Discrete Comput. Geom., 9, 4, 339-349 (1993) · Zbl 0784.52007 [122] Vre\'cica, Sini\v sa T.; \v Zivaljevi\'c, Rade T., New cases of the colored Tverberg theorem. Jerusalem combinatorics ’93, Contemp. Math. 178, 325-334 (1994), Amer. Math. Soc., Providence, RI · Zbl 0824.52010 [123] Wagner, U., On k-sets and applications (2003) [124] White, Moshe J., On Tverberg partitions, Israel J. Math., 219, 2, 549-553 (2017) · Zbl 1364.05012 [125] Whitney, Hassler, The self-intersections of a smooth $$n$$-manifold in $$2n$$-space, Ann. of Math. (2), 45, 220-246 (1944) · Zbl 0063.08237 [126] \v Zivaljevi\'c, Rade T., The Tverberg-Vre\'cica problem and the combinatorial geometry on vector bundles, Israel J. Math., 111, 53-76 (1999) · Zbl 0972.52005 [127] \v Zivaljevi\'c, Rade T.; Vre\'cica, Sini\v sa T., An extension of the ham sandwich theorem, Bull. London Math. Soc., 22, 2, 183-186 (1990) · Zbl 0709.60011 [128] \v Zivaljevi\'c, Rade T.; Vre\'cica, Sini\v sa T., The colored Tverberg’s problem and complexes of injective functions, J. Combin. Theory Ser. A, 61, 2, 309-318 (1992) · Zbl 0782.52003 [129] Zvagel’ski\u{\i}, M. Yu., An elementary proof of {T}verberg’s theorem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 353, Geometriya i Topologiya. 10, 54-61, 192 (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.