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Stromquist–Woodall Theorem
The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any ''n'' people with different tastes, and for any fraction ''w'', there exists a subset of the cake that all people value at exactly a fraction ''w'' of the total cake value, and it can be cut using at most 2n-2 cuts. The theorem is about a circular 1-dimensional cake (a "pie"). Formally, it can be described as the interval ,1in which the two endpoints are identified. There are ''n'' continuous measures over the cake: V_1,\ldots,V_n; each measure represents the valuations of a different person over subsets of the cake. The theorem says that, for every weight w \in ,1/math>, there is a subset C_w, which all people value at exactly w: : \forall i = 1,\ldots,n: \,\,\,\,\, V_i(C_w)=w, where C_w is a union of at most n-1 intervals. This means that 2n-2 cuts are sufficient for cutting the subset C_w. If the cake is not circular (that is, the endpoints are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics (especially social choice theory), dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods. The archetypal fair division algorithm is divide and choose. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Tukey 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 each one of them in half (with respect to their measure, e.g. volume) with a single -dimensional hyperplane. This is even possible if the objects overlap. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without taking the trouble to state the theorem in the -dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey. Naming The ham sandwich theorem takes its name from the case when and the three objects to be bisected are the ingredients of a ham sandwich. Sources differ on whether these three ingredients are two slices of bread and a piece of ham , bread and cheese and ham , or bread and butter and ham . In two dimensions, the theorem is known as the pancake theorem to refer to the flat nature of the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Pie-cutting
The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular. As an example, consider a birthday cake shaped as a disk. The cake should be divided among several children such that no child envies another child (as in a standard cake-cutting problem), with the additional constraint that the cuts must be radial, so that each child receives a circular sector. A possible application of the pie model might be for dividing an island’s shoreline into connected lots. Another possible application is in division of periodic time, such as dividing a daily cycle into "on-call" periods. Model A pie is usually modeled as the 1-dimensional interval ,2π(or ,1, in which the two endpoints are identified. This model was introduced in 1985 and later in 1993. Every procedure for fair cake-cutting can also be applied to cutting a pie by just ignoring the fact that the two endpoints are identified. For example, if the cake-cuttin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Exact division, also called consensus division, is a partition of a continuous resource (" cake") into some ''k'' pieces, such that each of ''n'' people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with ''k''=3 and ''n''=2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms: * Consensus halving – the cake should be partitioned into two pieces (''k''=2), and all agents agree that the pieces have equal values. *Consensus 1/''k''-division, for any constant ''k''>1 - the cake should be partition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake-cutting
Cake-cutting may refer to: * Fair cake-cutting, a mathematical problem of fairly dividing a heterogenous resource among people with different preferences ** Efficient cake-cutting Efficient cake-cutting is a problem in economics and computer science. It involves a ''heterogeneous'' resource, such as a cake with different toppings or a land with different coverings, that is assumed to be ''divisible'' - it is possible to cut a ..., a similar division problem in economics and computer science * Wedding-cake cutting, the habit of cutting the wedding cake and distributing it to the guests, as a symbol of fertility {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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