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83-454: KKM may refer to: Knaster–Kuratowski–Mazurkiewicz lemma , in mathematics Kyo Kara Maoh! , a series of Japanese light novels Kola Kolaya Mundhirika , a 2010 Tamil comedy film KKM , a song by the music group Miracle Legion from their 1996 album Portrait of a Damaged Family Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

166-450: A ( d + 1 ) {\displaystyle (d+1)} -tuple of points is a simplex ; in the plane it is a triangle and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of X {\displaystyle X} belongs to a simplex whose vertices belong to X {\displaystyle X} , and the third and fourth definitions are equivalent. In two dimensions,

249-504: A Euclidean space is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a given set X {\displaystyle X} may be defined as For bounded sets in the Euclidean plane, not all on one line, the boundary of the convex hull is the simple closed curve with minimum perimeter containing X {\displaystyle X} . One may imagine stretching

332-621: A convex polygon when d = 2 {\displaystyle d=2} , or more generally a convex polytope in R d {\displaystyle \mathbb {R} ^{d}} . Each extreme point of the hull is called a vertex , and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to S {\displaystyle S} and that encloses all of S {\displaystyle S} . For sets of points in general position ,

415-411: A rubber band so that it surrounds the entire set S {\displaystyle S} and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of S {\displaystyle S} . This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a spanning tree of

498-602: A "balanced collection" is. A collection B {\displaystyle B} of subsets of [ n ] {\displaystyle [n]} is called balanced if there is a weight function on B {\displaystyle B} (assigning a weight w J ≥ 0 {\displaystyle w_{J}\geq 0} to every J ∈ B {\displaystyle J\in B} ), such that, for each element i ∈ [ n ] {\displaystyle i\in [n]} ,

581-490: A KKM covering contains n closed sets, a KKMS covering contains 2 n − 1 {\displaystyle 2^{n}-1} closed sets - indexed by the nonempty subsets of [ n ] {\displaystyle [n]} (equivalently: by nonempty faces of Δ n − 1 {\displaystyle \Delta _{n-1}} ). For any I ⊆ [ n ] {\displaystyle I\subseteq [n]} ,

664-510: A connector. Any KKM covering is a connector-free covering, since in a KKM covering, no C i {\displaystyle C_{i}} even touches all n faces. However, there are connector-free coverings that are not KKM coverings. An example is illustrated at the right. There, the red set touches all three faces, but it does not contain any connector, since no connected component of it touches all three faces. A theorem of Ravindra Bapat , generalizing Sperner's lemma , implies

747-413: A convex combination of pure states in multiple ways. A convex hull in thermodynamics was identified by Josiah Willard Gibbs (1873), although the paper was published before the convex hull was so named. In a set of energies of several stoichiometries of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to

830-457: A convex hull whose boundary forms a continuously differentiable curve . However, for any angle θ {\displaystyle \theta } in the range π / 2 < θ < π {\displaystyle \pi /2<\theta <\pi } , there will be times during the Brownian motion where the moving particle touches the boundary of

913-526: A different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions. In the Arrow–Debreu model of general economic equilibrium , agents are assumed to have convex budget sets and convex preferences . These assumptions of convexity in economics can be used to prove

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996-413: A function f {\displaystyle f} on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of f {\displaystyle f} . It is the unique maximal convex function majorized by f {\displaystyle f} . The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of

1079-465: A hierarchical description of a given polygon called its convex differences tree . Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates. The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having

1162-463: A list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull. For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of n {\displaystyle n} ,

1245-453: A permutation π {\displaystyle \pi } of the coverings with a non-empty intersection, i.e: The name "rainbow KKM lemma" is inspired by Gale's description of his lemma: "A colloquial statement of this result is... if each of three people paint a triangle red, white and blue according to the KKM rules, then there will be a point which is in the red set of one person,

1328-404: A rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points . The convex hull operator is an example of a closure operator , and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding

1411-456: A set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination. For instance: The Delaunay triangulation of a point set and its dual , the Voronoi diagram , are mathematically related to convex hulls:

1494-437: A set of points undergoing insertions and deletions of points, and kinetic convex hull structures can keep track of the convex hull for points moving continuously. The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the rotating calipers method for computing the width and diameter of a point set. Several other shapes can be defined from

1577-529: A step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull. The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point). The convex hull of a finite point set S ⊂ R d {\displaystyle S\subset \mathbb {R} ^{d}} forms

1660-479: A unique minimal convex set containing X {\displaystyle X} , for every X {\displaystyle X} ? However, the second definition, the intersection of all convex sets containing X {\displaystyle X} , is well-defined. It is a subset of every other convex set Y {\displaystyle Y} that contains X {\displaystyle X} , because Y {\displaystyle Y}

1743-444: Is a balanced collection B ⊆ Faces ( P ) {\displaystyle B\subseteq {\textrm {Faces}}(P)} , such that the intersection of sets indexed by B {\displaystyle B} is nonempty: Komiya's theorem also generalizes the definition of a balanced collection: instead of requiring that there is a weight function on B {\displaystyle B} such that

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1826-404: Is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed. Outliers can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, or in

1909-628: Is a family of closed sets { C F : F ∈ Faces ( P ) } {\displaystyle \{C_{F}:F\in {\textrm {Faces}}(P)\}} such that for every face F ∈ Faces ( P ) {\displaystyle F\in {\textrm {Faces}}(P)} : F ⊆ ⋃ G ⊆ F ,   G ∈ Faces ( P ) C G . F\subseteq \bigcup _{G\subseteq F,~G\in {\textrm {Faces}}(P)}C_{G}. Komiya's theorem says that for every Komiya covering of P , there

1992-399: Is a special case of a KKMS covering. In a KKM covering, the n sets corresponding to singletons are nonempty, while the other sets are empty. However, there are many other KKMS coverings. in general, it is not true that the common intersection of all 2 n − 1 {\displaystyle 2^{n}-1} sets in a KKMS covering is nonempty; this is illustrated by

2075-454: Is called balanced with respect to b {\displaystyle {\textbf {b}}} iff b P ∈ conv ⁡ { b F : F ∈ B } {\displaystyle b^{P}\in \operatorname {conv} \{b^{F}:F\in B\}} , that is, the point assigned to the entire polygon P is a convex combination of the points assigned to

2158-466: Is generalized by the Krein–Smulian theorem , according to which the closed convex hull of a weakly compact subset of a Banach space (a subset that is compact under the weak topology ) is weakly compact. An extreme point of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set. For a convex hull, every extreme point must be part of

2241-419: Is illustrated by the picture on the right, in which set #1 is blue, set #2 is red and set #3 is green. The KKM requirements are satisfied, since: The KKM lemma states that there is a point covered by all three colors simultaneously; such a point is clearly visible in the picture. Note that it is important that all sets are closed, i.e., contain their boundary. If, for example, the red set is not closed, then it

2324-420: Is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing X {\displaystyle X} . Therefore, the first two definitions are equivalent. Each convex set containing X {\displaystyle X} must (by the assumption that it is convex) contain all convex combinations of points in X {\displaystyle X} , so

2407-418: Is possible that the central point is contained only in the blue and green sets, and then the intersection of all three sets may be empty. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to

2490-487: Is the barycenter of Δ n − 1 {\displaystyle \Delta _{n-1}} , which is the point ( 1 / n , … , 1 / n ) {\displaystyle (1/n,\ldots ,1/n)} ). Oleg R. Musin proved several generalizations of the KKM lemma and KKMS theorem, with boundary conditions on the coverings. The boundary conditions are related to homotopy . Convex hull In geometry ,

2573-472: Is the union of some of the features of the Delaunay triangulation, selected by comparing their circumradius to the parameter alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. The convex layers of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from

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2656-733: The Brouwer fixed-point theorem . Let Δ n − 1 {\displaystyle \Delta _{n-1}} be an ( n − 1 ) {\displaystyle (n-1)} -dimensional simplex with n vertices labeled as 1 , … , n {\displaystyle 1,\ldots ,n} . A KKM covering is defined as a set C 1 , … , C n {\displaystyle C_{1},\ldots ,C_{n}} of closed sets such that for any I ⊆ { 1 , … , n } {\displaystyle I\subseteq \{1,\ldots ,n\}} ,

2739-465: The convex hull of the vertices corresponding to I {\displaystyle I} is covered by ⋃ i ∈ I C i {\displaystyle \bigcup _{i\in I}C_{i}} . The KKM lemma says that in every KKM covering, the common intersection of all n sets is nonempty , i.e: When n = 3 {\displaystyle n=3} ,

2822-586: The convex hull of the vertices corresponding to I {\displaystyle I} should be covered by the union of sets corresponding to subsets of I {\displaystyle I} , that is: conv ⁡ ( { v i : i ∈ I } ) ⊆ ⋃ J ⊆ I C J {\displaystyle \operatorname {conv} (\{v_{i}:i\in I\})\subseteq \bigcup _{J\subseteq I}C_{J}} . Any KKM covering

2905-419: The convex hull , convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space , or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by

2988-403: The geometrization conjecture in low-dimensional topology . Hyperbolic convex hulls have also been used as part of the calculation of canonical triangulations of hyperbolic manifolds , and applied to determine the equivalence of knots . See also the section on Brownian motion for the application of convex hulls to this subject, and the section on space curves for their application to

3071-435: The local convex hull method by combining convex hulls of neighborhoods of points. In quantum physics , the state space of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are positive-semidefinite operators known as pure states and whose interior points are called mixed states. The Schrödinger–HJW theorem proves that any mixed state can in fact be written as

3154-507: The upper bound theorem in higher dimensions. As well as for finite point sets, convex hulls have also been studied for simple polygons , Brownian motion , space curves , and epigraphs of functions . Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the orthogonal convex hull , convex layers , Delaunay triangulation and Voronoi diagram , and convex skull . A set of points in

3237-454: The Delaunay triangulation of a point set in R n {\displaystyle \mathbb {R} ^{n}} can be viewed as the projection of a convex hull in R n + 1 . {\displaystyle \mathbb {R} ^{n+1}.} The alpha shapes of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. Each of alpha shape

3320-544: The KKM lemma considers the simplex Δ 2 {\displaystyle \Delta _{2}} which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets C 1 , C 2 , C 3 {\displaystyle C_{1},C_{2},C_{3}} such that: The KKM lemma states that the sets C 1 , C 2 , C 3 {\displaystyle C_{1},C_{2},C_{3}} have at least one point in common. The lemma

3403-418: The KKM lemma extends to connector-free coverings (he proved his theorem for n = 3 {\displaystyle n=3} ). The connector-free variant also has a permutation variant, so that both these generalizations can be used simultaneously. The KKMS theorem is a generalization of the KKM lemma by Lloyd Shapley . It is useful in economics , especially in cooperative game theory . While

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3486-441: The KKM lemma. Suppose we have a KKM covering C i {\displaystyle C_{i}} , for i = 1 , … , n {\displaystyle i=1,\ldots ,n} . Construct a KKMS covering C J ′ {\displaystyle C'_{J}} as follows: The KKM condition on the original covering C i {\displaystyle C_{i}} implies

3569-501: The KKMS condition on the new covering C J ′ {\displaystyle C'_{J}} . Therefore, there exists a balanced collection such that the corresponding sets in the new covering have nonempty intersection. But the only possible balanced collection is the collection of all singletons; hence, the original covering has nonempty intersection. The KKMS theorem has various proofs. Reny and Wooders proved that

3652-450: The balanced set can also be chosen to be partnered . Zhou proved a variant of the KKMS theorem where the covering consists of open sets rather than closed sets. Hidetoshi Komiya generalized the KKMS theorem from simplices to polytopes . Let P be any compact convex polytope. Let Faces ( P ) {\displaystyle {\textrm {Faces}}(P)} be the set of nonempty faces of P . A Komiya covering of P

3735-416: The convex hull at a point of angle θ {\displaystyle \theta } . The Hausdorff dimension of this set of exceptional times is (with high probability) 1 − π / 2 θ {\displaystyle 1-\pi /2\theta } . For the convex hull of a space curve or finite set of space curves in general position in three-dimensional space,

3818-445: The convex hull is a simplicial polytope . According to the upper bound theorem , the number of faces of the convex hull of n {\displaystyle n} points in d {\displaystyle d} -dimensional Euclidean space is O ( n ⌊ d / 2 ⌋ ) {\displaystyle O(n^{\lfloor d/2\rfloor })} . In particular, in two and three dimensions

3901-410: The convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points (points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls,

3984-411: The convex hull of X {\displaystyle X} is already a closed set itself (as happens, for instance, if X {\displaystyle X} is a finite set or more generally a compact set ), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way. If

4067-431: The convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces , are fundamental problems of computational geometry . They can be solved in time O ( n log ⁡ n ) {\displaystyle O(n\log n)} for two or three dimensional point sets, and in time matching the worst-case output complexity given by

4150-431: The convex hull of an open set is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed. For instance, the closed set (the set of points that lie on or above the witch of Agnesi ) has the open upper half-plane as its convex hull. The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces,

4233-405: The convex hull of its eigenvalues . The Russo–Dye theorem describes the convex hulls of unitary elements in a C*-algebra . In discrete geometry , both Radon's theorem and Tverberg's theorem concern the existence of partitions of point sets into subsets with intersecting convex hulls. The definitions of a convex set as containing line segments between its points, and of a convex hull as

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4316-402: The convex hull property Bézier curves helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the home range . Newton polygons of univariate polynomials and Newton polytopes of multivariate polynomials are convex hulls of points derived from the exponents of the terms in

4399-558: The existence of an equilibrium. When actual economic data is non-convex , it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market. In geometric modeling , one of the key properties of a Bézier curve is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves. In

4482-392: The faces in the collection B . The KKMS theorem is a special case of Komiya's theorem in which the polytope P = Δ n − 1 {\displaystyle P=\Delta _{n-1}} and b F {\displaystyle b^{F}} is the barycenter of the face F (in particular, b P {\displaystyle b^{P}}

4565-429: The geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the hull of the vessel. It differs from the skin girth , the perimeter of the cross-section itself, except for boats and ships that have a convex hull. The convex hull is commonly known as the minimum convex polygon in ethology , the study of animal behavior, where it

4648-456: The given set, because otherwise it cannot be formed as a convex combination of given points. According to the Krein–Milman theorem , every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space ) is the convex hull of its extreme points. However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and

4731-694: The hull, its distance to the new hull represents the degree of stability of the phase. The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. The term "convex hull" itself appears as early as the work of Garrett Birkhoff  ( 1935 ), and the corresponding term in German appears earlier, for instance in Hans Rademacher 's review of Kőnig  ( 1922 ). Other terms, such as "convex envelope", were also used in this time frame. By 1938, according to Lloyd Dines ,

4814-508: The intersection of all convex supersets, apply to hyperbolic spaces as well as to Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of ideal points , points that do not belong to the hyperbolic space itself but lie on the boundary of a model of that space. The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to ruled surfaces in Euclidean space, and their metric properties play an important role in

4897-415: The number of faces is at most linear in n {\displaystyle n} . The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets . Computing the same decomposition recursively for each pocket forms

4980-410: The number of input points, and h {\displaystyle h} , the number of points on the convex hull, which may be significantly smaller than n {\displaystyle n} . For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis. Graham scan can compute the convex hull of n {\displaystyle n} points in

5063-412: The objects. The definition using intersections of convex sets may be extended to non-Euclidean geometry , and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces ; convex hulls may also be generalized in a more abstract way, to oriented matroids . It is not obvious that the first definition makes sense: why should there exist

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5146-530: The open convex hull of a set X {\displaystyle X} is d {\displaystyle d} -dimensional, then every point of the hull belongs to an open convex hull of at most 2 d {\displaystyle 2d} points of X {\displaystyle X} . The sets of vertices of a square, regular octahedron, or higher-dimensional cross-polytope provide examples where exactly 2 d {\displaystyle 2d} points are needed. Topologically,

5229-403: The open unit ball are both convex, but neither one has any extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces. The convex-hull operator has the characteristic properties of a closure operator : When applied to a finite set of points, this is the closure operator of an antimatroid ,

5312-580: The other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column. David Gale proved the following generalization of the KKM lemma. Suppose that, instead of one KKM covering, we have n different KKM coverings: C 1 1 , … , C n 1 , … , C 1 n , … , C n n {\displaystyle C_{1}^{1},\ldots ,C_{n}^{1},\ldots ,C_{1}^{n},\ldots ,C_{n}^{n}} . Then, there exists

5395-438: The outermost contour of Tukey depth , are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules . Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics . In economics, convex hulls can be used to apply methods of convexity in economics to non-convex markets. In geometric modeling,

5478-524: The parts of the boundary away from the curves are developable and ruled surfaces . Examples include the oloid , the convex hull of two circles in perpendicular planes, each passing through the other's center, the sphericon , the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter. The convex hull or lower convex envelope of

5561-557: The plane in time O ( n log ⁡ n ) {\displaystyle O(n\log n)} . For points in two and three dimensions, more complicated output-sensitive algorithms are known that compute the convex hull in time O ( n log ⁡ h ) {\displaystyle O(n\log h)} . These include Chan's algorithm and the Kirkpatrick–Seidel algorithm . For dimensions d > 3 {\displaystyle d>3} ,

5644-416: The points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the obstacle problem of finding a minimum-energy surface above a given shape can have the convex hull as their solution. For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of

5727-492: The points that are not vertices of the convex hull. The convex skull of a polygon is the largest convex polygon contained inside it. It can be found in polynomial time , but the exponent of the algorithm is high. Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials , matrix eigenvalues , and unitary elements , and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as

5810-452: The polynomial, and can be used to analyze the asymptotic behavior of the polynomial and the valuations of its roots. Convex hulls and polynomials also come together in the Gauss–Lucas theorem , according to which the roots of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial. In spectral analysis , the numerical range of a normal matrix is

5893-532: The risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules. In combinatorial optimization and polyhedral combinatorics , central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions. In multi-objective optimization ,

5976-529: The second and third definitions are equivalent. In fact, according to Carathéodory's theorem , if X {\displaystyle X} is a subset of a d {\displaystyle d} -dimensional Euclidean space, every convex combination of finitely many points from X {\displaystyle X} is also a convex combination of at most d + 1 {\displaystyle d+1} points in X {\displaystyle X} . The set of convex combinations of

6059-400: The set of all convex combinations is contained in the intersection of all convex sets containing X {\displaystyle X} . Conversely, the set of all convex combinations is itself a convex set containing X {\displaystyle X} , so it also contains the intersection of all convex sets containing X {\displaystyle X} , and therefore

6142-537: The shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension. The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides

6225-414: The special case of a KKM covering, in which most sets are empty. The KKMS theorem says that, in every KKMS covering, there is a balanced collection B {\displaystyle B} of 2 [ n ] {\displaystyle 2^{[n]}} , such that the intersection of sets indexed by B {\displaystyle B} is nonempty : It remains to explain what

6308-503: The sum of weights near each vertex of P is 1, we start by choosing any set of points b = { b F : F ∈ Faces ( P ) , b F ∈ F } {\displaystyle {\textbf {b}}=\{b^{F}:F\in {\textrm {Faces}}(P),b^{F}\in F\}} . A collection B ⊆ Faces ( P ) {\displaystyle B\subseteq {\textrm {Faces}}(P)}

6391-407: The sum of weights of all subsets containing i {\displaystyle i} is exactly 1. For example, suppose n = 3 {\displaystyle n=3} . Then: In hypergraph terminology , a collection B is balanced with respect to its ground-set V , iff the hypergraph with vertex-set V and edge-set B admits a perfect fractional matching. The KKMS theorem implies

6474-433: The theory of developable surfaces . In robust statistics , the convex hull provides one of the key components of a bagplot , a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth. In statistical decision theory ,

6557-415: The time for computing the convex hull is O ( n ⌊ d / 2 ⌋ ) {\displaystyle O(n^{\lfloor d/2\rfloor })} , matching the worst-case output complexity of the problem. The convex hull of a simple polygon in the plane can be constructed in linear time . Dynamic convex hull data structures can be used to keep track of the convex hull of

6640-553: The title KKM . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=KKM&oldid=847989463 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Knaster%E2%80%93Kuratowski%E2%80%93Mazurkiewicz lemma The KKM lemma can be proved from Sperner's lemma and can be used to prove

6723-502: The union of their epigraphs, or equivalently from their pointwise minimum ) and, in this form, is dual to the convex conjugate operation. In computational geometry , a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. Output representations that have been considered for convex hulls of point sets include

6806-451: The upward-facing and downward-facing parts of the boundary form topological disks. The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior ) of the convex hull. The closed convex hull of X {\displaystyle X} is the intersection of all closed half-spaces containing X {\displaystyle X} . If

6889-583: The white set of another, the blue of the third". The rainbow KKM lemma can be proved using a rainbow generalization of Sperner's lemma . The original KKM lemma follows from the rainbow KKM lemma by simply picking n identical coverings. A connector of a simplex is a connected set that touches all n faces of the simplex. A connector-free covering is a covering C 1 , … , C n {\displaystyle C_{1},\ldots ,C_{n}} in which no C i {\displaystyle C_{i}} contains

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