Papers
Topics
Authors
Recent
Search
2000 character limit reached

Towards the mathematical foundation of the minimum enclosing ball and related problems

Published 9 Jan 2024 in cs.CG, cs.AI, and math.GT | (2402.06629v1)

Abstract: Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (118)
  1. Euclid (author), Hypsicles of Alexandria (contributor), Giovanni Mocenigo (dedicatee), Euclid’s “Elements”. Venice: Erhard Ratdolt, May 25, 1482. [Pdf] Retrieved from the Library of Congress, [Website] https://www.loc.gov/item/2021667076/.
  2. J. J. Sylvester, “A question in the geometry of situation,” Quart. J. Pure Appl. Math., vol. 1, p. 79, 1857.
  3. J. J. Sylvester, “On Poncelet’s approximate valuation of surd forms,” Philosophical Magazine, vol. 120, pp. 203–222, 1860.
  4. G. Chrystal, “On the problem to construct the minimum circle enclosing n𝑛nitalic_n given points in a plane,” Proc. Edinburgh Math. Soc., vol. 3, pp. 30–33, Third Meeting, Jan. 1885.
  5. H. W. E. Jung, “Über die kleinste Kugel, die eine räumliche Figur einschliesst,” J. Reine Angew. Math., vol. 123, no. 3, pp. 241–257, 1901.
  6. P. Steinhagen, “Über die größte Kugel in einer konvexen Punktmenge,” Abh. Math. Semin. Univ. Hambg., vol. 1, no. 1, pp. 15–26, 1921.
  7. N. Alon, S. Dar, M. Parnas, and D. Ron, “Testing of clustering,” SIAM Review, vol. 46, no. 2, pp. 285–308, 2004.
  8. M. Bădoiu, S. Har-Peled, and P. Indyk, “Approximate clustering via core-sets,” in Thirty-fourth Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 250–257, Montreal, Quebec, Canada: ACM, 2002.
  9. S. Chakraborty, R. Pratap, S. Roy, and S. Saraf, “Helly-type theorems in property testing,” Internat. J. Comput. Geom. Appl., vol. 28, no. 4, pp. 365–379, 2018.
  10. I. Bárány and J. Kalai, “Helly-type problems,” Bull. Amer. Math. Soc. (N.S.), vol. 59, no. 4, pp. 471–502, 2022.
  11. L. M. Blumenthal and G. E. Wahlin, “On the spherical surface of smallest radius enclosing a bounded subset of n𝑛nitalic_n-dimensional Euclidean space,” Bull. Amer. Math. Soc., vol. 47, pp. 771–777, Feb. 1941.
  12. J. H. C. Whitehead, “On C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-complexes,” Ann. of Math. (2), vol. 41, no. 4, pp. 809–824, 1940.
  13. M. N. Vrahatis, “Generalization of the Apollonius theorem for simplices and related problems,” 2024, arXiv:2401.03232.
  14. E. Welzl, “Smallest enclosing disks (balls and ellipsoids),” in New Results and New Trends in Computer Science, H. Maurer (ed), Lect. Notes Comput. Sci., vol. 555, pp. 359–370, Berlin: Springer, 1991.
  15. F. Nielsen and G. Hadjeres, “Approximating covering and minimum enclosing balls in hyperbolic geometry,” in Geometric Science of Information, (GSI 2015), F. Nielsen and F. Barbaresco (eds), Lect. Notes Comput. Sci., vol. 9389, pp. 586–594, Cham, Switzerland: Springer 2015.
  16. M. Arnaudon and F. Nielsen, “On approximating the Riemannian 1-center,” Comput. Geom. Theory Appl., vol. 46, no. 1, pp. 93–104, Jan. 2013.
  17. F. Nielsen, “The Siegel-Klein disk: Hilbert geometry of the Siegel disk domain,” Entropy, vol. 22, no. 9, art. no. 1019, pp. 1–41, Sep. 2020.
  18. A. Mazumdar, Y. Polyanskiy, and B. Saha, “On Chebyshev radius of a set in Hamming space and the closest string problem,” in Proc. International Symposium on Information Theory (ISIT 2013), pp. 1401–1405, Istanbul, Turkey: IEEE 2013.
  19. R. Friederich, A. Ghosh, M. Graham, B. Hicks, and R. Shevchenko, “Experiments with unit disk cover algorithms for covering massive pointsets,” Comput. Geom. Theory Appl., vol. 109, 101925, pp. 1–24, 2023.
  20. B. Mordukhovich, N. M. Nam, and C. Villalobos, “The smallest enclosing ball problem and the smallest intersecting ball problem: Existence and uniqueness of solutions,” Optim. Lett., vol. 7, pp. 839–853, Jun. 2013.
  21. H. Ding, “Stability yields sublinear time algorithms for geometric optimization in machine learning,” in 29th Annual European Symposium on Algorithms (ESA 2021), P. Mutzel, R. Pagh, and G. Herman (eds), vol. 204 of LIPIcs, Article No. 38, pp. 38:1–38:19, Leibniz-Zentrum für Informatik, Wadern, Germany: Dagstuhl Publishing, 2021.
  22. M. Cavaleiro and F. Alizadeh, “A branch-and-bound method for the minimum k𝑘kitalic_k-enclosing ball problem,” Oper. Res. Lett., vol. 50, no. 3, pp. 274–280, 2022.
  23. V. V. Shenmaier, “The problem of a minimal ball enclosing k𝑘kitalic_k points,” J. Appl. Ind. Math., vol. 7, no. 3, pp. 444–448, 2013.
  24. S. D. Ahipaşaoğlu and E. A. Yıldırım “Identification and elimination of interior points for the minimum enclosing ball problem,” SIAM J. Optim., vol. 19, no. 3, pp. 1392–1396, 2008.
  25. K. Fischer and B. Gärtner, “The smallest enclosing ball of balls: Combinatorial structure and algorithms,” Int. J. Comput. Geom. Appl., vol. 14, no. 4-5, pp. 341–378, 2004.
  26. S. G. Kolliopoulos and S. Rao, “A nearly linear-time approximation scheme for the Euclidean k𝑘kitalic_k-median problem,” SIAM J. Comput., vol. 37, no. 3, pp. 757–782, 2007.
  27. A. Bhattacharya, D. Goyal, and R. Jaiswal, “Hardness of approximation for Euclidean k𝑘kitalic_k-median,” in Approximation, Randomization, and Combinatorial Optimization Algorithms and Techniques (APPROX/RANDOM 2021), Leibniz International Proceedings in Informatics (LIPIcs), vol. 207, pp. 4:1-4:23, Leibniz-Zentrum für Informatik, Wadern, Germany: Dagstuhl Publishing, 2021.
  28. V. Verter, “Uncapacitated and capacitated facility location problems,” in Foundations of Location Analysis, H. A. Eiselt and V. Marianov (eds.), International Series in Operations Research & Management Science, vol. 155, ch. 2, pp. 25-37, Springer, 2011.
  29. S. G. Kolliopoulos and Y. Moysoglou, “Integrality gaps for strengthened linear relaxations of capacitated facility location,” Math. Program., vol. 158, no. 1-2, pp. 99–141, 2016.
  30. S. Zhang, “Challenges in KNN classification,” IEEE Trans. Knowl. Data Eng., vol. 34, no. 10, pp. 4663–4675, Oct. 2022.
  31. P. K. Syriopoulos, N. G. Kalampalikis, S. B. Kotsiantis, and M. N. Vrahatis, “kNN classification: a review,” Ann. Math. Artif. Intell., in press, 2023.
  32. P. Indyk and A. Naor, “Nearest-neighbor-preserving embeddings,” ACM Trans. Algorithms, vol. 3, no. 3, Art. 31, pp. 1–12, 2007.
  33. I. Z. Emiris, V. Margonis, and I. Psarros, “Near-neighbor preserving dimension reduction via coverings for doubling subsets of ℓ1subscriptℓ1{\ell}_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT,” Theor. Comput. Sci., vol. 942, pp. 169–179, 2023.
  34. N. T. An, N. M. Nam, and X. Qin, “Solving k𝑘kitalic_k-center problems involving sets based on optimization techniques,” J. Glob. Optim., vol. 76, pp. 189–209, 2020.
  35. A. Banik, B. B. Bhattacharya, and S. Das, “Minimum enclosing circle of a set of fixed points and a mobile point,” Comput. Geom. Theory Appl., vol. 47, no. 9, pp. 891–898, Oct. 2014.
  36. S. Roy, A. Karmakar, S. Das, and S. C. Nandy, “Constrained minimum enclosing circle with center on a query line segment,” Comput. Geom. Theory Appl., vol. 42, no. 6-7, pp. 632–638, Aug. 2009.
  37. M. Imanparast, S. N. Hashemi, and A. Mohades, “Efficient approximation algorithms for point-set diameter in higher dimensions,” J. Algorithms Comput., vol. 51, no. 2 pp. 47–61, Dec. 2019.
  38. T. M. Chan, “Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus,” Int. J. Comput. Geom. Appl., vol. 12, no. 1-2, pp. 67–85, 2002.
  39. C. A. Duncan, M. T. Goodrich, and E. A. Ramos, “Efficient approximation and optimization algorithms for computational metrology,” in Proc. Eighteth Annual Symposium on Discrete Algorithms (SODA 1997), pp. 121–130, New Orleans, LA: ACM/SIAM, Jan. 1997.
  40. P. K. Agarwal, B. Aronov, S. Har-Peled, and M. Sharir, “Approximation algorithms for minimum-width annuli and shells,” Discrete Comput. Geom., vol. 24, no. 4, pp. 687–705, 2000.
  41. B. Aronov and M. Dulieu, “How to cover a point set with a V-shape of minimum width,” Comput. Geom. Theory Appl., vol. 46, no. 3, pp. 298–309, 2013.
  42. T. M. Chan and S. Har-Peled, “Smallest k𝑘kitalic_k-enclosing rectangle revisited,” Discrete Comput. Geom., vol. 66, no. 2, pp. 769–791, 2021.
  43. S. Z. Selim and M. A. Ismail, “K𝐾Kitalic_K-means-type algorithms: A generalized convergence theorem and characterization of local optimality,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-6, no. 1, pp. 81–87, Jan. 1984.
  44. A. Vattani, “k𝑘kitalic_k-means requires exponentially many iterations even in the plane,” Discrete Comput. Geom., vol. 45, no. 4, pp. 596–616, 2011.
  45. M. N. Vrahatis, B. Boutsinas, P. Alevizos, and G. Pavlides, “The new k𝑘kitalic_k-windows algorithm for improving the k𝑘kitalic_k-means clustering algorithm,” J. Complexity, vol. 18, no. 1, pp. 375–391, 2002.
  46. S. W. Bae, “Computing a minimum-width square or rectangular annulus with outliers,” Comput. Geom. Theory Appl., vol. 76, no. 3, pp. 33–45, 2019.
  47. S. I. Veselov, D. V. Gribanov, and D. S. Malyshev, “FPT-algorithm for computing the width of a simplex given by a convex hull,” Moscow Univ. Comput. Math. Cybern., vol. 43, no. 1, pp. 1–11, 2019.
  48. L. G. Casado, I. García, B. G. Tóth, and E. M. T. Hendrix, “On determining the cover of a simplex by spheres centered at its vertices,” J. Glob. Optim., vol. 50, pp. 645–655, 2011.
  49. H. Everett, I. Stojmenovic, P. Valtr, and S. Whitesides, “The largest k𝑘kitalic_k-ball in a d𝑑ditalic_d-dimensional box,” Comput. Geom. Theory Appl., vol. 11, no. 2, pp. 59–67, 1998.
  50. A. Kanazawa, “On the minimal volume of simplices enclosing a convex body,” Arch. Math. (Basel), vol. 102, no. 5, pp. 489–492, 2014.
  51. D. Galicer, M. Merzbacher, and D. Pinasco, “The minimal volume of simplices containing a convex body,” J. Geom. Anal., vol. 29, no. 1, pp. 717–32, 2019.
  52. G. Ya. Perel′′{}^{\prime}\!start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPTman, “k𝑘kitalic_k-radii of a convex body,” Sibirsk. Mat. Zh., vol. 28, no. 4, pp. 185–186, 1987.
  53. S. V. Pukhov “Inequalities for the Kolmogorov and Bernšteĭn widths in Hilbert space,” Mat. Zametki, vol. 25, no. 4, pp. 619–628, 637, 1979.
  54. B. González Merino, “Improving bounds for the Perel′′{}^{\prime}\!start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPTman-Pukhov quotient for inner and outer radii,” J. Convex Anal., vol. 24, no. 4, pp. 1099–1116, 2017.
  55. F. Plastria, “Continuous covering location problems,” in Facility Location: Application and Theory, Z. Drezner and H. W. Hamacher (eds), Chapter 2, pp. 37–79, New York, USA: Springer, 2002.
  56. Y. Zhou and J. Zhu, “FocusPaka/MEB: A Minimum Enclosing Ball (MEB) method to detect differential expression genes for RNA-seq data,” GitHub packages, License: GPL-2, Version: 0.99.4, built on October 22, 2022, [Website] https://rdrr.io/github/FocusPaka/MEB/
  57. F. A. Elerian, W. M. K. Helal, and M. A. AbouEleaz, “Methods of roundness measurement: An experimental comparative study,” J. Mech. Eng. Res. Dev., vol. 44, no. 9, pp. 173–183, 2021.
  58. C. J. C. Burges, “A tutorial on support vector machines for pattern recognition,” Data Min. Knowl. Discov., vol. 2, no. 2, pp. 121–167, 1998.
  59. A. Ben-Hur, D. Horn, H. T. Siegelmann, and V. Vapnik, “Support vector clustering,” J. Mach. Learn. Res., vol. 2, pp. 125–137, 2001.
  60. Y. Bulatov, S. Jambawalikar, P. Kumar, and S. Sethia, “Hand recognition using geometric classifiers,” in International Conference on Biometric Authentication (ICBA 2004), D. Zhang and A. K. Jain (eds), Lect. Notes Comput. Sci., vol. 3072, pp. 753–759, Berlin, Heidelberg: Springer, 2004.
  61. N. Alon, S. Dar, M. Parnas, and D. Ron, “Testing of clustering,” in Forty-first Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 240–250, Los Alamitos, CA: IEEE, 2000.
  62. N. Alon, S. Dar, M. Parnas, and D. Ron, “Testing of clustering,” SIAM J. Discrete Math., vol. 16, no. 3, pp. 393–417, 2003.
  63. A. Rizwan, N. Iqbal, R. Ahmad, and D.-H. Kim, “WR-SVM model based on the margin radius approach for solving the minimum enclosing ball problem in support vector machine classification,” Appl. Sci., vol. 11, no. 10, 4657, pp. 1–21, Jun. 2021.
  64. O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee, “Choosing multiple parameters for support vector machines,” Mach. Learn., vol. 46, no. 1/3, pp. 131–159, 2002.
  65. Y. Ren and F. Zhang, “Hand gesture recognition based on MEB-SVM,” in Proc. International Conference on Embedded Software and Systems, Hangzhou, China, 2009, pp. 344–349.
  66. A. Goel, P. Indyk, and K. R. Varadarajan, “Reductions among high dimensional proximity problems,” in Proc. Twelfth Annual Symposium on Discrete Algorithms (SODA 2001), S. Rao Kosaraju (ed), pp. 769–778, Washington, DC, USA: ACM/SIAM, 2001.
  67. P. M. Hubbard, “Approximating polyhedra with spheres for time-critical collision detection,” ACM Trans. Graph., vol. 15, no. 3, pp. 179–210, 1996.
  68. T. Larsson, G. Capannini, and L. Källberg, “Parallel computation of optimal enclosing balls by iterative orthant scan,” Comput. Graph., vol. 56, pp. 1–10, May 2016.
  69. C. L. Lawson, “The smallest covering cone or sphere (Problem 63-5* by C. Groenewoud and L. Eusanio),” SIAM Review, vol. 7, no. 3, pp. 415–416, 1965.
  70. M. Tzelepi and A. Tefas, “Improving the performance of light-weight CNN models using minimum enclosing ball regularization,” in Proc. 27th European Signal Processing Conference (EUSIPCO 2019), pp. 1–5, A Coruña, Spain: IEEE 2019.
  71. R. Kurniawati, J. S. Jin, and J. A. Shepherd, “SS+{}^{+}start_FLOATSUPERSCRIPT + end_FLOATSUPERSCRIPT–tree: An improved index structure for similarity searches in a high-dimensional feature space,” in 5th Storage and Retrieval for Image and Video Databases, I. K. Sethi and R. C. Jain (eds), vol. 3022, pp. 110–120, San Jose, CA, USA: SPIE, 1997.
  72. F.-L. Chung, Z. H. Deng, and S. T. Wang, “From minimum enclosing ball to fast fuzzy inference system training on large datasets,” IEEE Trans. Fuzzy Syst., vol. 17, no. 1, pp. 173–184, Feb. 2009.
  73. J. M. Abowd, “The US Census Bureau adopts differential privacy,” in Proc. 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2018), 2018, pp. 2867–2867.
  74. B. Ghazi, R. Kumar, and P. Manurangsi, “Differentially private clustering: Tight approximation ratios,” in Proc. Thirty-Fourth Conference on Neural Information Processing Systems (NeurIPS 2020), art. no. 340, pp. 4040–4054, Vancouver, Canada, December 2020.
  75. I. W.-H. Tsang, J. T.-Y. Kwok, and P.-M. Cheung, “Core vector machines: Fast SVM training on very large data sets,” J. Mach. Learn. Res., vol. 6, no. 13, pp. 363–392, 2005.
  76. I. W.-H. Tsang, J. T.-Y. Kwok, and J. M. Zurada, “Generalized core vector machines,” IEEE Trans. Neural Netw., vol. 17, no. 5, pp. 1126–1140, Sep. 2006.
  77. F. Nielsen and R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, 2009.
  78. S. Har-Peled and E. Robson, “On the width of the regular n𝑛nitalic_n-simplex,” 2023, arXiv:2301.02616.
  79. R. M. Tanner, “Some content maximizing properties of the regular simplex,” Pacific J. Math., vol. 52, no. 2, pp. 611–616, 1974.
  80. R. Alexander, “The width and diameter of a simplex,” Geometriae Dedicata, vol. 6, no. 1, pp. 87–94, 1977.
  81. P. Gritzmann and V. Klee, “Inner and outer j𝑗jitalic_j-radii of convex bodies in finite-dimensional normed spaces,” Discrete Comput. Geom., vol. 7, no. 3, pp. 255–280, 1992.
  82. W. Blaschke, “Über den grössten Kreis in einer konvexen Punktmenge,” Jahresber. Dtsch. Math.-Ver., vol. 23, pp. 369–374, 1914.
  83. R. Brandenberg, “Radii of regular polytopes,” Discrete Comput. Geom., vol. 33, no. 1, pp. 43–55, 2005.
  84. H. W. E. Jung, “Über den kleinsten Kreis, der eine ebene Figur einschliesst,” J. Reine Angew. Math., vol. 137, no. 4, pp. 310–313, 1910.
  85. E. Landau, “Über einige Ungleichheitsbeziehungen in der Theorie der analytischen Funktionen,” Arch. Math. Phys., vol. 11, no. 3, pp. 31–36, 1905.
  86. W. Süss, “Durchmesser und Umkugel bei mehrdimensionalen Punktmengen,” Math. Z., vol. 40, no. 1, pp. 315–316, 1936.
  87. F. Bohnenblust, “Convex regions and projections in Minkowski spaces,” Ann. Math., vol. 39, no. 2, pp. 301–308, 1938.
  88. K. Leichtweiss, “Zwei Extremalprobleme der Minkowski – Geometrie,” Math. Zeitschr., vol. 62, pp. 37–49, 1955.
  89. J. Daneš, “On the radius of a set in a Hilbert space,” Commentat. Math. Univ. Carolin., vol. 25, no. 2, pp. 355–362, 1984.
  90. M. Katz, “Jung’s theorem in complex projective geometry,” Quart. J. Math. Oxford Ser. (2), vol. 36, no. 144, pp. 451–466, 1985.
  91. B. V. Dekster, “An extension of Jung’s theorem,” Israel J. Math., vol. 50, no. 3, pp. 169–180, Sep. 1985.
  92. B. V. Dekster, “The Jung theorem for the spherical and the hyperbolic spaces,” Acta Math. Hungar., vol. 67, no. 4, pp. 315–331, Dec. 1995.
  93. B. V. Dekster, “The Jung theorem in metric spaces of curvature bounded above,” Proc. Amer. Math. Soc., vol. 125, no. 8, pp. 2425–2433, Aug. 1997.
  94. U. Lang and V. Schroeder, “Jung’s theorem for Alexandrov spaces of curvature bounded above,” Ann. Glob. Anal. Geom., vol. 15, no. 3, pp. 263–275, 1997.
  95. V. Nguen-Khac and K. Nguen-Van, “An infinite-dimensional generalization of the Jung theorem,” Math. Notes, vol. 80, no. 2, pp. 224–232, 2006.
  96. V. Boltyanski and H. Martini, “Jung’s theorem for a pair of Minkowski spaces,” Adv. Geom., vol. 6, no. 4, pp. 645–650, Oct. 2006.
  97. A. V. Akopyan, “Combinatorial generalizations of Jung’s theorem,” Discrete Comput. Geom., vol. 49, no. 3, pp. 478–484, Mar. 2013.
  98. R. Brandenberg and B. González Merino, “The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant,” Israel J. Math., vol. 218, no. 1, pp. 489–510, Mar. 2017.
  99. H. Gericke, “Über die größte Kugel in einer konvexen Punktmenge,” Math. Z., vol. 40, pp. 317–320, 1936.
  100. L. A. Santaló, “Convex regions on the n𝑛nitalic_n-dimensional spherical surface,” Ann. Math., vol. 47, no. 3, pp. 448–459, Jul. 1946.
  101. L. A. Santaló, “Propiedades de las figuras convexas sobre la esfera,” Mathematicae Notae, vol. 4, pp. 11–40, 1944.
  102. R. M. Robinson, “Note on convex regions on the sphere,” Bull. Amer. Math. Soc., vol. 44, pp. 115–116, Feb. 1938.
  103. M. Henk, “A generalization of Jung’s theorem,” Geom. Dedicata, vol. 42, pp. 235–240, 1992.
  104. U. Betke and M. Henk, “A generalization of Steinhagen’s theorem,” Abh. Math. Semin. Univ. Hambg., vol. 63, no. 1, pp. 165–176, 1993.
  105. D. Gale, “On inscribing n𝑛nitalic_n-dimensional sets in a regular n𝑛nitalic_n-simplex,” Proc. Amer. Math. Soc., vol. 4, no. 2, pp. 222–225, 1953.
  106. M. N. Vrahatis, “An error estimation for the method of bisection in ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT,” Bull. Greek Math. Soc., vol. 27, pp. 161–174, 1986.
  107. M. N. Vrahatis, “A variant of Jung’s theorem,” Bull. Greek Math. Soc., vol. 29, pp. 1–6, 1988.
  108. C. Carathéodory, “Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen,” Math. Ann., vol. 64, no. 1, pp. 95–115, 1907.
  109. E. Helly, “Über Mengen konvexer Körper mit gemeinschaftlichen Punkten,” Jahresber. Dtsch. Math.-Ver., vol. 32, pp. 175–176, 1923.
  110. H. A. Tverberg, “A generalization of Radon’s theorem,” J. London Math. Soc., vol. 41, pp. 123–128, 1966.
  111. J. A. De Loera, X. Goaoc, F. Meunier, and N. Mustafa, “The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg,” Bull. Amer. Math. Soc. (N.S.), vol. 56, no. 3, pp. 415–511, 2019.
  112. L. Danzer, B. Grünbaum, and V. Klee, “Helly’s theorem and its relatives,” in Convexity, Proc. Symp. Pure Math., vol. 7, Amer. Math. Soc., Providence, RI, 1963, pp. 101–180.
  113. H. A. Tverberg, “A generalization of Radon’s theorem II,” Bull. Austral. Math. Soc., vol. 24, no. 3, pp. 321–325, 1981.
  114. J. Radon, “Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten,” Math. Ann., vol. 83, no. 1-2, pp. 113–115, 1921.
  115. B. J. Birch, “On 3⁢N3𝑁3N3 italic_N points in a plane,” Proc. Cambridge Philos. Soc., vol. 55, pp. 289–293, 1959.
  116. K. Adiprasito, I. Bárány, N. Mustafa, and T. Terpai, “Theorems of Carathéodory, Helly, and Tverberg without dimension,” Discrete Comput. Geom., vol. 64, no. 2, pp. 233–258, 2020.
  117. I. Bárány, “A generalization of Carathéodory’s theorem,” Discrete Math., vol. 40, no. 2-3, pp. 141–152, 1982.
  118. I. V. Polikanova and G. Ya. Perel′′{}^{\prime}start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPTman, “A remark on Helly’s theorem,” Sibirsk. Mat. Zh., vol. 27, no. 5, pp. 191–194, 207, 1986.
Citations (4)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (1)

  1. Michael N. Vrahatis 

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up