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Theorem about the intersections of d-dimensional convex sets
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913, but not published
Helly's_theorem
On convergent subsequences of functions that are locally of bounded total variation
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions
Helly's_selection_theorem
Family of sets where every disjoint subfamily has k or fewer sets
s_{n}\neq \emptyset } . These concepts are named after Eduard Helly (1884–1943); Helly's theorem on convex sets, which gave rise to this notion, states that
Helly_family
Theorem in geometry about convex sets
In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that: Any set of d + 2 points in Rd can be partitioned into two
Radon's_theorem
Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle
Doignon's_theorem
Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set
Kirchberger's_theorem
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
Two other theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Theorem in probability theory
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain
Helly–Bray_theorem
On convergent subsequences of regulated functions
In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of
Fraňková–Helly selection theorem
Fraňková–Helly_selection_theorem
Austrian mathematician (1884–1943)
Eduard Helly (June 1, 1884 in Vienna – 28 November 1943 in Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem
Eduard_Helly
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Area of functional analysis and convex analysis
Carathéodory's theorem – Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P Helly's theorem – Theorem about the intersections
Choquet_theory
Theorem on extension of bounded linear functionals
the theorem for the space C [ a , b ] {\displaystyle C[a,b]} of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and
Hahn–Banach_theorem
Russian mathematician (born 1966)
V. Polikanova, he established a measure-theoretic formulation of Helly's theorem.[PP86] In 1987, the year he began graduate studies, he published an
Grigori_Perelman
Multivariate generalization of the median
simple proof of the existence of a centerpoint may be obtained using Helly's theorem. Suppose there are n points, and consider the family of closed half-spaces
Centerpoint_(geometry)
Proof all ranked voting rules have spoilers
Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no procedure for group
Arrow's_impossibility_theorem
Set of strings with few differences
searching within this Hamming ball to find the solution. A version of Helly's theorem for Hamming balls is known: For Hamming balls of radius r {\displaystyle
Hamming_ball
About simultaneous modular congruences
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then
Chinese_remainder_theorem
Polygon that is the boundary of a convex set
consisting in adding diagonals from one vertex to all other vertices. Helly's theorem: For every collection of at least three convex polygons: if all intersections
Convex_polygon
Upper bound on intersecting set families
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common.
Erdős–Ko–Rado_theorem
On decreasing nested sequences of non-empty compact sets
Cantor's intersection theorem, also called Cantor's nested intervals theorem, refers to two closely related theorems in general topology and real analysis
Cantor's_intersection_theorem
On when a space equals the closed convex hull of its extreme points
Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex
Krein–Milman_theorem
American computer scientist
from the University of California, Berkeley in 1994. Her dissertation, Helly Theorems and Generalized Linear Programming, was supervised by Raimund Seidel
Nina_Amenta
In geometry, set whose intersection with every line is a single line segment
fixed-point theorem Complex convexity Convex cone Convex series Convex metric space Carathéodory's theorem (convex hull) Choquet theory Helly's theorem Holomorphically
Convex_set
Any collection of sets, or subsets of a set
intersection has bounded size. Helly's theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families. An abstract simplicial
Family_of_sets
Mathematical optimization problem restricted to integers
ISSN 0364-765X. Amenta, Nina; De Loera, Jesús A.; Soberón, Pablo (2017). "Helly's theorem: new variations and applications". In Harrington, Heather A.; Omar
Integer_programming
Mathematical method
selection Helly's selection theorem Zero-dimensional Michael selection theorem Robert Aumann measurable selection theorem Blaschke selection theorem Maximum
Selection_theorem
Mathematics of convex functions and sets
Carathéodory's theorem (convex hull), Helly's theorem, and Radon's theorem describe the combinatorial and geometric structure of convex hulls. These theorems are
Convex_analysis
On when a family of real, continuous functions has a uniformly convergent subsequence
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence
Arzelà–Ascoli_theorem
Combinatorial game theory theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap
Sprague–Grundy_theorem
each side of it. The existence of a centerpoint can be proved using Helly's theorem. For a given point p and constant a>0, define Pr(a,p,o) as the probability
Geometric_separator
Theorem in functional analysis
and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of
Banach–Alaoglu_theorem
Book on discrete geometry
every two points, then the given points must all lie on a single line. Helly's theorem, that if a family of compact convex sets has a non-empty intersection
Combinatorial Geometry in the Plane
Combinatorial_Geometry_in_the_Plane
Abstraction of ordered linear algebra
Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma of Farkas—can be formulated
Oriented_matroid
the original body. Hadwiger's theorem - a theorem that characterizes the valuations on convex bodies in Rn. Helly's theorem Hyperplane - a subspace whose
List_of_convexity_topics
In board games that cannot end in a draw, one of the two players has a winning strategy
In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which
Zermelo's theorem (game theory)
Zermelo's_theorem_(game_theory)
On partitions into intersecting convex hulls
In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in Euclidean space can be
Tverberg's_theorem
Class of theorems about Nash equilibrium payoff profiles in repeated games
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971). The
Folk_theorem_(game_theory)
Solution concept of a non-cooperative game
Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose
Nash_equilibrium
Gives condition for a set of functions to be relatively compact in an Lp space
In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition
Fréchet–Kolmogorov_theorem
Theorem in game theory
Aumann's agreement theorem states that two Bayesian agents with the same prior beliefs cannot "agree to disagree" about the probability of an event if
Aumann's_agreement_theorem
measure. By a compactness argument (or equivalently in this case Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures
Positive_harmonic_function
German mathematician (1896–1981)
known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the Siegel
Carl_Ludwig_Siegel
Mixed strategy equilibria explained as the limit of pure strategy equilibria
In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem justifies a puzzling aspect of mixed strategy
Purification_theorem
Pairing where no unchosen pair prefers each other over their choice
and hybrid CPU–GPU execution to reduce overhead. The rural hospitals theorem concerns a more general variant of the stable matching problem, like that
Stable_matching_problem
Logical paradox in decision-making theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Paradox_of_tolerance
Weakly optimal allocation of resources
per the Greenwald–Stiglitz theorem. The second welfare theorem is essentially the reverse of the first welfare theorem. It states that under similar
Pareto_efficiency
Hungarian and American mathematician and physicist (1903–1957)
the application of this work was instrumental in his mean ergodic theorem. The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle
John_von_Neumann
continuity theorem Darmois–Skitovich theorem Edgeworth series Helly–Bray theorem Kac–Bernstein theorem Location parameter Maxwell's theorem Moment-generating
List_of_probability_topics
Subfield of set theory
This fact—that all closed games are determined—is called the Gale–Stewart theorem. Note that by symmetry, all open games are determined as well. (A game
Determinacy
Field of economics and game theory
described by Noam Nisan as a way to escape the Gibbard–Satterthwaite theorem. While the theorem is traditionally presented as a result about voting systems, it
Mechanism_design
Academic discipline
often-cited paper describing experiments which could be used to prove Bell's theorem. In one part of this paper, they describe a game where a player could have
Quantum_game_theory
Paper-and-pencil game for two players
successful landing and must be careful not to block themself. Hales–Jewett theorem m,n,k-game Number Scrabble Garcia, Dan. "GamesCrafters: Tic-Tac-Toe". gamescrafters
Tic-tac-toe
Normed vector space that is complete
(1998) Theorem 1.12.11, p. 112 in Megginson (1998) Theorem 2.5.16, p. 216 in Megginson (1998). see II.A.8, p. 29 in Wojtaszczyk (1991) see Theorem 2.6.23
Banach_space
Principle in economics and game theory
system or market). It can be seen as a kind of mirror image to Gibbard's theorem. The revelation principle says that if a social choice function can be
Revelation_principle
Decision rule used for minimizing the possible loss for a worst-case scenario
important in the theory of repeated games. One of the central theorems in this theory, the folk theorem, relies on the minimax values. In combinatorial game theory
Minimax
Problem about bus travel
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Wait/walk_dilemma
Interval graph, proper Line graph Lollipop graph Minor Robertson–Seymour theorem Pairwise compatibility graph Petersen graph Planar graph Dual polyhedron
List_of_graph_theory_topics
Hungarian mathematician (1866–1942)
functions of a complex variable: he is the eponym of an important class of theorems with applications ranging from mathematical and harmonic analysis to number
Alfred_Tauber
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Strategic_move
Game-theoretic concept
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Cheap_talk
Concept in game theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Focal_point_(game_theory)
Smallest convex set containing a given set
Bárány, Imre; Katchalski, Meir; Pach, János (1982), "Quantitative Helly-type theorems", Proceedings of the American Mathematical Society, 86 (1): 109–114
Convex_hull
Military strategy during the Cold War with regard to the use of nuclear weapons
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Deterrence_theory
Making of satisfactory, not optimal, decisions
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Bounded_rationality
Situation where total gains match total losses
non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium
Zero-sum_game
Concept in game theory
ranked voting with three or more alternatives (by the Gibbard–Satterthwaite theorem) or first-price auctions. A randomized mechanism is a probability-distribution
Incentive_compatibility
Standard example in game theory
Abilene paradox Centipede game Collective action problem Externality Folk theorem (game theory) Free-rider problem Gift-exchange game Hobbesian trap Innocent
Prisoner's_dilemma
Mathematical models of strategic interactions
von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard
Game_theory
Concept in game theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Shapley_value
Search algorithm
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Alpha–beta_pruning
Polish-American mathematician (1911-1992)
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Melvin_Dresher
Solution to the fair division problem
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Moving-knife_procedure
Two player pursuit-evasion problem
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Princess_and_monster_game
σ-finite measure, but the converse is again not true. Helly's selection theorem Helly–Bray theorem Klenke, Achim (2008). Probability Theory. Berlin: Springer
Sub-probability_measure
English saying meaning "equivalent retaliation"
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Tit_for_tat
Hand game for two or more players
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Chopsticks_(hand_game)
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Manipulated_Nash_equilibrium
Solution concept in game theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Proper_equilibrium
Simple mathematical game
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Pirate_game
Game whose outcome can be correctly predicted
Computer Go Computer Othello Game complexity God's algorithm Zermelo's theorem (game theory) Allis, L.V. (1994). Searching for solutions in games and
Solved_game
Austrian mathematician (1865–1945)
Gebilde im Gebiete von mehreren komplexen Veränderlichen" [An integral theorem on analytic forms on a domain of several complex variables], Monatshefte
Wilhelm_Wirtinger
Diplomatic policy of concessions
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Appeasement
Hand game for two players or more
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Rock_paper_scissors
Term in game theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Move_by_nature
Game theory scenario
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Win–win_game
Concept in conflict studies
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Conflict_escalation
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Complete_mixing
Solution concept in game theory
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Quasi-perfect_equilibrium
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Symmetric_equilibrium
Israeli-American psychologist and economist (1934–2024)
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Daniel_Kahneman
Type of perfect Bayesian equilibrium
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Separating_equilibrium
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
List_of_games_in_game_theory
Graph with a median for each three vertices
of lattice operations and inequalities is Theorem 1 of Birkhoff & Kiss (1947). Birkhoff & Kiss (1947), Theorem 2. Birkhoff & Kiss (1947), p. 751. Avann
Median_graph
Trigger strategy
defection. Brinkmanship – Political and military tactic Folk theorem (game theory) – Class of theorems about Nash equilibrium payoff profiles in repeated games
Grim_trigger
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Fictitious_play
Economic model
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Stackelberg_competition
attrition Theorems Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite
Asynchrony_(game_theory)
Set in game theory
inequalities. Hence the core is closed and convex. The Bondareva–Shapley theorem: the core of a game is nonempty if and only if the game is "balanced".
Core_(game_theory)
Class of strategies employed in a repeated non-cooperative game
A non-cooperative equilibrium for supergames, Review of Economic Studies 38, 1–12. (The first formal proof of the Folk theorem (game theory)). v t e
Trigger_strategy
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