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Ham sandwich theorem

Index Ham sandwich theorem

In mathematical measure theory, for every positive integer the ham sandwich theorem states that given measurable "objects" in -dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single -dimensional hyperplane. [1]

49 relations: Algorithm, American Mathematical Monthly, Antipodal point, Arthur Harold Stone, Big O notation, Borsuk–Ulam theorem, Bread, Computational geometry, Constantin Carathéodory, Continuous function, Counting measure, Dimension, Discrete & Computational Geometry, Discrete geometry, Duke Mathematical Journal, Equivariant topology, Euclidean space, Euclidean vector, Exact division, Fair cake-cutting, Finite set, Function (mathematics), Ham, Hugo Steinhaus, Hyperplane, Intermediate value theorem, John Tukey, Line (geometry), Linear combination, Linear continuum, Linear independence, Mathematics, Measure (mathematics), N-sphere, Normal (geometry), Orientation (geometry), Origin (mathematics), Outer measure, Pancake, Plane (geometry), Point (geometry), Robertson–Webb rotating-knife procedure, Sandwich, Sphere, Squeeze theorem, Stanislaw Ulam, Stefan Banach, Subset, Unit sphere.

Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter.

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Arthur Harold Stone

Arthur Harold Stone (30 September 1916 – 6 August 2000) was a British mathematician born in London, who worked mostly in topology.

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Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

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Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point.

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Bread

Bread is a staple food prepared from a dough of flour and water, usually by baking.

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Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry.

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Constantin Carathéodory

Constantin Carathéodory (Greek: Κωνσταντίνος Καραθεοδωρή Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Counting measure

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Discrete & Computational Geometry

Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer.

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Discrete geometry

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.

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Duke Mathematical Journal

Duke Mathematical Journal is a peer-reviewed mathematics journal published by Duke University Press.

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Equivariant topology

In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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Exact division

An exact division, also called even division or consensus division, is a division of a heterogeneous resource ("cake") to several subsets such that each of n people with different tastes agree about the valuations of the pieces.

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Fair cake-cutting

Fair cake-cutting is a kind of fair division problem.

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Finite set

In mathematics, a finite set is a set that has a finite number of elements.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Ham

Ham is pork from a leg cut that has been preserved by wet or dry curing, with or without smoking.

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Hugo Steinhaus

Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Jewish-Polish mathematician and educator.

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Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

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John Tukey

John Wilder Tukey (June 16, 1915 – July 26, 2000) was an American mathematician best known for development of the FFT algorithm and box plot.

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Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear continuum

In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

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N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

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Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

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Orientation (geometry)

In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies.

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Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

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Outer measure

In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions.

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Pancake

A pancake (or hotcake, griddlecake, or flapjack) is a flat cake, often thin and round, prepared from a starch-based batter that may contain eggs, milk and butter and cooked on a hot surface such as a griddle or frying pan, often frying with oil or butter.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

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Robertson–Webb rotating-knife procedure

The Robertson–Webb rotating-knife procedure is a procedure for envy-free cake-cutting of a two-dimensional cake among three partners.

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Sandwich

A sandwich is a food typically consisting of vegetables, sliced cheese or meat, placed on or between slices of bread, or more generally any dish wherein two or more pieces of bread serve as a container or wrapper for another food type.

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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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Squeeze theorem

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.

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Stanislaw Ulam

Stanisław Marcin Ulam (13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics.

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Stefan Banach

Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.

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Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.

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Bisection theorem, Ham sandwich cut, Ham sandwich problem, Ham-sandwich cut, Ham-sandwich theorem, Pancake theorem, Stone-Tukey theorem, Stone–Tukey theorem.

References

[1] https://en.wikipedia.org/wiki/Ham_sandwich_theorem

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