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We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry.
Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen
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Bedingt konvergente Reihen und konvexe Systeme. i-ii-iii
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Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten
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Über einige Anwendungen diophantischer Approximationen
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Komplexe in Euklidischen Räumen
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Über die Existenz n n -dimensionaler Komplexe, die nicht in den ℝ 2 n \mathbb{R}^{2n} topologisch einbettbar sind
Flores, A. I · 1933
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Méthodes topologiques dans les problèmes variationnels
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On extreme points of regular convex sets
M. Krein, D. M · 1940
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Generalized “sandwich” theorems
Stone, A., and Tukey, J · 1942
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Theory of Games and Economic Behavior
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On sets of distances of n n points
Erdős, P · 1946
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A theorem on general measure
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On the imbedding of systems of compacta in simplicial complexes
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Sur la division pragmatique
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Equilibrium points in n n -person games
Nash, Jr., J. F · 1950
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Non-cooperative games
Nash, Jr., J. F · 1951
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A generalization of Tucker’s combinatorial lemma with topological applications
Fan, K · 1952
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Some games and machines for playing them
Nash, Jr., J. F · 1952
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The double description method
Motzkin, T. S., Raiffa, H., Thompson, G. L., and Thrall, R · 1953
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Solutions of irreflexive relations
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On a proof of Helly’s theorem on sets of convex bodies with common points
Krasnosel ′ · 1954
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Constructive proof of the Min-Max theorem
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On 3 N 3N points in a plane
Birch, B · 1959
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How to cut a cake fairly
Dubins, L. E., and Spanier, E. H · 1961
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On convergence proofs on perceptrons
Novikoff, A. B · 1962
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Helly’s theorem and its relatives
Danzer, L., Grünbaum, B., and Klee, V · 1963
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Equilibrium points of bimatrix games
Lemke, C. E., and Howson, Jr., J. T · 1964
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Generalizations of a theorem of Carathéodory
Reay, J · 1965
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A generalization of Radon’s theorem
Tverberg, H · 1966
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Some intersection properties of convex bodies
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Simplicial maps from an orientable n n -pseudomanifold into S m {S}^{m} with the octahedral triangulation
Fan, K · 1967
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Neuere Probleme der Kombinatorik
Ryser, H. J · 1967
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The approximation of fixed points of a continuous mapping
Scarf, H. E · 1967
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The core of an N N person game
Scarf, H. E · 1967
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Intersection and covering properties of convex sets
Chakerian, G. D · 1969
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Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers
Kay, D., and Womble, E · 1971
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On the uniform convergence of relative frequencies of events to their probabilities
Vapnik, V. N., and Chervonenkis, A. Y · 1971
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How good is the simplex algorithm?
Klee, V., and Minty, G · 1972
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Normal hypergraphs and the perfect graph conjecture
Lovász, L · 1972
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On the computational complexity of finding a kernel
Chvátal, V · 1973
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Convexity in cristallographical lattices
Doignon, J.-P · 1973
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Subjectivity and correlation in randomized strategies
Aumann, R · 1974
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On the foundations of linear and integer linear programming I
Graver, J. E · 1975
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Transversals of Latin squares and their generalizations
Stein, S. K · 1975
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Mathematics and the picturing of data
Tukey, J · 1975
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On division of graphs to connected subgraphs
Győri, E · 1976
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n n -tuple colorings and associated graphs
Stahl, S · 1976
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A theorem concerning the integer lattice
Bell, D · 1977
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A min-max relation for submodular functions on graphs
Edmonds, J., and Giles, R · 1977
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An observation on the structure of production sets with indivisibilities
Scarf, H. E · 1977
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The number of necessary constraints in an integer program: a new proof of Scarf’s theorem
Todd, M. J · 1977
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Kneser’s conjecture, chromatic number, and homotopy
Lovász, L · 1978
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Vertex-critical subgraphs of Kneser graphs
Schrijver, A · 1978
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On a common generalization of Borsuk’s and Radon’s theorem
Bajmóczy, E. G., and Bárány, I · 1979
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Theoretical properties of the network simplex method
Cunningham, W. H · 1979
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The game of Hex and the Brouwer fixed-point theorem
Gale, D · 1979
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Total dual integrality and integer polyhedra
Giles, F., and Pulleyblank, W · 1979
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Binding constraints and Helly numbers
Hoffman, A. J · 1979
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A problem of geometry in ℝ n \mathbb{R}^{n}
Katchalski, M., and Liu, A · 1979
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Graphes noyau-parfaits
Duchet, P · 1980
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Polynomial algorithms in linear programming
Khachiyan, L. G · 1980
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Remarques sur un résultat non publié de B. Maurey
Pisier, G · 1980
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Decomposition of regular matroids
Seymour, P. D · 1980
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How to cut a cake fairly
Stromquist, W · 1980
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Dividing a cake fairly
Woodall, D. R · 1980
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On a topological generalization of a theorem of Tverberg
Bárány, I., Shlosman, S. B., and Szücs, A · 1981
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Transversals of families of sets
Dol’nikov, V · 1981
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A constructive proof of Tucker’s combinatorial lemma
Freund, R. M., and Todd, M. J · 1981
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A generalization of Radon’s theorem. II
Tverberg, H · 1981
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A generalization of Carathéodory’s theorem
Bárány, I · 1982
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Quantitative Helly-type theorems
Bárány, I., Katchalski, M., and Pach, J · 1982
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Seminar, MSHi (Maison des Sciences de l’Homme), 1983
Berge, C., and Duchet, P · 1983
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Simple proofs of some Borsuk-Ulam results
Dold, A · 1983
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Une généralisation du théorème de Richardson sur l’existence de noyaux dans les graphes orientés
Duchet, P., and Meyniel, H · 1983
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Descriptive statistics for multivariate distributions
Oja, H · 1983
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Extremal problems in discrete geometry
Szemerédi, E., and Trotter, W · 1983
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The number of triangles covering the center of an n n -set
Boros, E., and Füredi, Z · 1984
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Equilibrium in a discrete exchange economy with money
Gale, D · 1984
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On kernels and semikernels of digraphs
Galeana-Sánchez, H., and Neumann-Lara, V · 1984
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Intersection patterns of convex sets
Kalai, G · 1984
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A new polynomial-time algorithm for linear programming
Karmarkar, N · 1984
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Elements of algebraic topology
Munkres, J. R · 1984
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Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces
Carl, B · 1985
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An upper-bound theorem for families of convex sets
Eckhoff, J · 1985
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Bisection of circle colorings
Goldberg, C., and West, D · 1985
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A general approach to d d -dimensional geometric queries
Yao, A. C., and Yao, F. F · 1985
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The Borsuk-Ulam theorem and bissection of necklaces
Alon, N., and West, D · 1986
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An integer analogue of Carathéodory’s theorem
Cook, W., Fonlupt, J., and Schrijver, A · 1986
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Theory of linear and integer programming
Schrijver, A · 1986
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Splitting necklaces
Alon, N · 1987
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New applications of random sampling in computational geometry
Clarkson, K. L · 1987
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Convexity in combinatorial structures
Duchet, P · 1987
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ϵ \epsilon -nets and simplex range queries
Haussler, D., and Welzl, E · 1987
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Equivariant maps for the symmetric group
Özaydin, M · 1987
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On a notion of simplicial depth
Liu, R · 1988
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NP-completeness of the linear complementarity problem
Chung, S. J · 1989
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Applications of random sampling in computational geometry, II
Clarkson, K. L., and Shor, P. W · 1989
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Nash and correlated equilibria: Some complexity considerations
Gilboa, I., and Zemel, E · 1989
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Combinatorial complexity bounds for arrangement of curves and spheres
Clarkson, K. L., Edelsbrunner, H., Guibas, L., Sharir, M., and Welzl, E · 1990
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Separators in two and three dimensions
Miller, G. L., and Thurston, W. P · 1990
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Hilbert bases, Carathéodory’s theorem and combinatorial optimization
Sebő, A · 1990
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Combinatorial Matrix Theory
Brualdi, R. A., and Ryser, H. J · 1991
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On the geometry and computational complexity of Radon partitions in the integer lattice
Onn, S · 1991
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Small-dimensional linear programming and convex hulls made easy
Seidel, R · 1991
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Point selections and weak ϵ \epsilon -nets for convex hulls
Alon, N., Bárány, I., Füredi, Z., and Kleitman, D. J · 1992
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Piercing convex sets and the Hadwiger-Debrunner ( p , q ) (p,q) -problem
Alon, N., and Kleitman, D. J · 1992
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A colored version of Tverberg’s theorem
Bárány, I., and Larman, D. G · 1992
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Geometric medians
Gil, J., Steiger, W. L., and Wigderson, A · 1992
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A subexponential randomized simplex algorithm
Kalai, G · 1992
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Efficient partition trees
Matoušek, J · 1992
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Tverberg’s theorem via number fields
Sarkaria, K. S · 1992
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The colored Tverberg’s problem and complexes of injective functions
Živaljević, R. T., and Vrećica, S. T · 1992
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Geometric and combinatorial applications of Borsuk’s theorem
Bárány, I · 1993
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Cutting hyperplanes for divide-and-conquer
Chazelle, B · 1993
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Improved bounds on weak ϵ \epsilon -nets for convex sets
Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Sharir, M., and Welzl, E · 1993
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Helly, Radon, and Carathéodory type theorems
Eckhoff, J · 1993
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Handbook of Convex Geometry
Gruber, P., and Wills, J · 1993
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A new triangulation for simplicial algorithms
Todd, M. J., and Tunçel, L · 1993
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On generalizations of Radon’s theorem and the ham sandwich theorem
Tverberg, H., and Vrećica, S · 1993
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Theory of convex structures
van de Vel, M · 1993
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Note on a conjecture of Sierksma
Vučić, A., and Živaljević, R. T · 1993
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Helly theorems and generalized linear programming
Amenta, N · 1994
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Computing a centerpoint of a finite planar set of points in linear time
Jadhav, S., and Mukhopadhyay, A · 1994
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Algorithms for ham-sandwich cuts
Lo, C., Matoušek, J., and Steiger, W · 1994
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On the complexity of the parity argument and other inefficient proofs of existence
Papadimitriou, C. H · 1994
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Carathéodory’s theorem, colourful and applicable
Bárány, I., and Onn, S · 1995
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Topological methods
Björner, A · 1995
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Almost optimal set covers in finite VC-dimension
Brönnimann, H., and Goodrich, M · 1995
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Las Vegas algorithms for linear and integer programming when the dimension is small
Clarkson, K. L · 1995
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A condition for matchability of hypergraphs
Haxell, P. E · 1995
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Combinatorics and convexity
Kalai, G · 1995
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Combinatorial Geometry
Pach, J., and Agarwal, P. K · 1995
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Perfect graphs are kernel solvable
Boros, E., and Gurvich, V · 1996
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Fair Division: From cake-cutting to dispute resolution
Brams, S., and Taylor, A · 1996
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Approximating center points with iterative Radon points
Clarkson, K. L., Eppstein, D., Miller, G., Sturtivant, C., and Teng, S · 1996
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New lower bounds for Hopcroft’s problem
Erickson, J · 1996
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A subexponential bound for linear programming
Matoušek, J., Sharir, M., and Welzl, E · 1996
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Computation of equilibria in finite games
McKelvey, R. D., and McLennan, A · 1996
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Algorithm AS 307: Bivariate location depth
Rousseeuw, P., and Ruts, I · 1996
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Transversal numbers over subsets of linear spaces
Averkov, G., and Weismantel, R · 2012
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Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Chan, T. M., Grant, E., Könemann, J., and Sharpe, M · 2012
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Algorithmic solutions for envy-free cake cutting
Deng, X., Qi, Q., and Saberi, A · 2012
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Polyhedra with the integer Carathéodory property
Gijswijt, D., and Regts, G · 2012
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On the chromatic number of generalized stable Kneser graphs, 2012
Jonsson, J · 2012
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A simpler proof of the Boros-Füredi-Bárány-Pach-Gromov theorem
Karasev, R. N · 2012
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Colourful linear programming and its relatives
Bárány, I., and Onn, S · 1997
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The combinatorics of timetabling
de Werra, D · 1997
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A user’s guide to algebraic topology
Dodson, C. T. J., and Parker, P. E · 1997
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Graph homomorphisms: structure and symmetry
Hahn, G., and Tardif, C · 1997
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Fractional kernels in digraphs
Aharoni, R., and Holzman, R · 1998
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A new lower bound based on Gromov’s method of selecting heavily covered points
Král, D., Mach, L., and Sereni, J · 2012
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A generalisation of Tverberg’s theorem
Soberón, P., and Strausz, R · 2012
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The equivalence of linear programs and zero-sum games
Adler, I · 2013
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On maximal S-free sets and the Helly number for the family of S-convex sets
Averkov, G · 2013
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N N -person cake-cutting: there may be no perfect division
Brams, S. J., Jones, M. A., and Klamler, C · 2013
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Oja centers and centers of gravity
Chen, D., Devillers, O., Iacono, J., Langerman, S., and Morin, P · 2013
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Algebraic and Geometric Ideas in the Theory of Discrete Optimization
De Loera, J. A., Hemmecke, R., and Köppe, M · 2013
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A Course in Topological Combinatorics
de Longueville, M · 2013
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Envy-free two-player m -cake and three-player two-cake divisions
Lebert, N., Meunier, F., and Carbonneaux, Q · 2013
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A further generalization of the colourful Carathéodory theorem
Meunier, F., and Deza, A · 2013
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Algorithms for tolerated Tverberg partitions
Mulzer, W., and Stein, Y · 2013
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Approximating Tverberg points in linear time for any fixed dimension
Mulzer, W., and Werner, D · 2013
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A Borsuk-Ulam equivalent that directly implies Sperner’s lemma
Nyman, K. L., and Su, F. E · 2013
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Intersection patterns of convex sets via simplicial complexes: A survey
Tancer, M · 2013
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Implementation of a unimodularity test
Walter, M., and Truemper, K · 2013
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Proofs from THE BOOK
Aigner, M., and Ziegler, G. M · 2014
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Tverberg plus constraints
Blagojević, P. V. M., Frick, F., and Ziegler, G. M · 2014
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The mathematics of decisions, elections, and games
Crisman, K.-D., and Jones, M. A · 2014
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Helly numbers of acyclic families
De Verdière, É. C., Ginot, G., and Goaoc, X · 2014
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Low-crossing spanning trees: an alternative proof and experiments
Giannopoulos, P., Konzack, M., and Mulzer, W · 2014
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Polynomiality for bin packing with a constant number of item types
Goemans, M. X., and Rothvoß, T · 2014
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Matoušek, J., and Wagner, U · 2014
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A proof of the Oja depth conjecture in the plane
Mustafa, N. H., Tiwary, H. R., and Werner, D · 2014
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Lectures on stochastic programming. Modeling and theory
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Depth with respect to a family of convex sets
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Homotopy invariants of covers and KKM-type lemmas
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An optimal generalization of the colorful Carathéodory theorem
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Proof of a conjecture of Bárány, Katchalski and Pach
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Colorful subhypergraphs in uniform hypergraphs
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