Quantitative Tverberg, Helly, & Carathéodory theorems

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.

[1]  Jesús A. De Loera,et al.  A quantitative Doignon-Bell-Scarf theorem , 2014, Comb..

[2]  Márton Naszódi.  Proof of a Conjecture of Bárány, Katchalski and Pach , 2016, Discret. Comput. Geom..

[3]  J. D. Loera,et al.  Helly numbers of subsets of ℝᵈ and sampling techniques in optimization , 2015 .

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

[5]  Jesús A. De Loera,et al.  Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf's Theorem , 2014, IPCO.

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

[7]  Gennadiy Averkov,et al.  On Maximal S-Free Sets and the Helly Number for the Family of S-Convex Sets , 2011, SIAM J. Discret. Math..

[8]  A. Barvinok.  Thrifty approximations of convex bodies by polytopes , 2012, 1206.3993.

[9]  Günter M. Ziegler.  3N Colored Points in a Plane , 2011 .

[10]  Robert Weismantel,et al.  Transversal numbers over subsets of linear spaces , 2010, 1002.0948.

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

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

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

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

[15]  E. Bronstein.  Approximation of convex sets by polytopes , 2008 .

[16]  Rephael Wenger,et al.  Helly-Type Theorems and Geometric Transversals , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[17]  I. Bárány.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[18]  Jiří Matoušek,et al.  A fractional Helly theorem for convex lattice sets , 2003 .

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

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

[21]  Shlomo Reisner,et al.  Dropping a vertex or a facet from a convex polytope , 2001 .

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

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

[24]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[25]  Y. Gordon,et al.  Umbrellas and Polytopal Approximation of the Euclidean Ball , 1996, math/9603208.

[26]  Y. Gordon,et al.  Constructing a polytope to approximate a convex body , 1995 .

[27]  Kenneth L. Clarkson,et al.  Las Vegas algorithms for linear and integer programming when the dimension is small , 1995, JACM.

[28]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

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

[30]  P. Gruber.  Aspects of Approximation of Convex Bodies , 1993 .

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

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

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

[34]  Stanley Rabinowitz,et al.  A Theorem about Collinear Lattice Points , 1989 .

[35]  J. Pach,et al.  Quantitative Helly-type theorems , 1982 .

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

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

[38]  Gerard Sierksma,et al.  A Tverberg-type generalization of the Helly number of a convexity space , 1981 .

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

[40]  A. J. Hoffman.  BINDING CONSTRAINTS AND HELLY NUMBERS , 1979 .

[41]  H. Scarf.  An observation on the structure of production sets with indivisibilities. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

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

[43]  E. M. Bronshteyn,et al.  The approximation of convex sets by polyhedra , 1975 .

[44]  R. Dudley.  Metric Entropy of Some Classes of Sets with Differentiable Boundaries , 1974 .

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

[46]  D. C. Kay,et al.  Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers , 1971 .

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

[48]  V. Klee,et al.  Helly's theorem and its relatives , 1963 .

[49]  K. Strubecker.  Affine Differentialgeometrie , 1936 .

[50]  E. Helly,et al.  Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten , 1930 .

[51]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

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

[53]  E. Steinitz.  Bedingt konvergente Reihen und konvexe Systeme. , 1913 .

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