Convex hull

This article is about the smallest convex shape enclosing a given shape. For boats whose hulls are convex, seeHull (watercraft) § Hull shapes.

Ingeometry,theconvex hull,convex envelopeorconvex closureof a shape is the smallestconvex setthat contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of aEuclidean space,or equivalently as the set of allconvex combinationsof points in the subset. For aboundedsubset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

The convex hull of the red set is the blue and redconvex set.

Convex hulls ofopen setsare open, and convex hulls ofcompact setsare compact. Every compact convex set is the convex hull of itsextreme points.The convex hull operator is an example of aclosure operator,and everyantimatroidcan be represented by applying this closure operator to finite sets of points. Thealgorithmicproblems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and itsdualproblem of intersectinghalf-spaces,are fundamental problems ofcomputational geometry.They can be solved in time$O(n\log n)$for two or three dimensional point sets, and in time matching the worst-case output complexity given by theupper bound theoremin higher dimensions.

As well as for finite point sets, convex hulls have also been studied forsimple polygons,Brownian motion,space curves,andepigraphs of functions.Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include theorthogonal convex hull,convex layers,Delaunay triangulationandVoronoi diagram,andconvex skull.

Definitions

Convex hull of a bounded planar set: rubber band analogy

A set of points in aEuclidean spaceis defined to beconvexif it contains the line segments connecting each pair of its points. The convex hull of a given set$X$may be defined as

1. The (unique) minimal convex set containing$X$
2. The intersection of all convex sets containing$X$
3. The set of allconvex combinationsof points in$X$
4. The union of allsimpliceswith vertices in$X$

Forbounded setsin the Euclidean plane, not all on one line, the boundary of the convex hull is thesimple closed curvewith minimumperimetercontaining$X$.One may imagine stretching arubber bandso that it surrounds the entire set$S$and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of$S$.This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of aspanning treeof the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull.However, in higher dimensions, variants of theobstacle problemof 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 convexbounding volumeof the objects. The definition using intersections of convex sets may be extended tonon-Euclidean geometry,and the definition using convex combinations may be extended from Euclidean spaces to arbitraryreal vector spacesoraffine spaces;convex hulls may also be generalized in a more abstract way, tooriented matroids.

Equivalence of definitions

3D convex hull of 120 point cloud

It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing$X$,for every$X$?However, the second definition, the intersection of all convex sets containing$X$,is well-defined. It is a subset of every other convex set$Y$that contains$X$,because$Y$is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing$X$.Therefore, the first two definitions are equivalent.

Each convex set containing$X$must (by the assumption that it is convex) contain all convex combinations of points in$X$,so the set of all convex combinations is contained in the intersection of all convex sets containing$X$.Conversely, the set of all convex combinations is itself a convex set containing$X$,so it also contains the intersection of all convex sets containing$X$,and therefore the second and third definitions are equivalent.

In fact, according toCarathéodory's theorem,if$X$is a subset of a$d$-dimensional Euclidean space, every convex combination of finitely many points from$X$is also a convex combination of at most$d+1$points in$X$.The set of convex combinations of a$(d+1)$-tuple of points is asimplex;in the plane it is atriangleand in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of$X$belongs to a simplex whose vertices belong to$X$,and the third and fourth definitions are equivalent.

Upper and lower hulls

In two dimensions, 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, the upward-facing and downward-facing parts of the boundary form topological disks.

Topological properties

Closed and open hulls

Theclosed convex hullof a set is theclosureof the convex hull, and theopen convex hullis theinterior(or in some sources therelative interior) of the convex hull.

The closed convex hull of$X$is the intersection of all closedhalf-spacescontaining$X$. If the convex hull of$X$is already aclosed setitself (as happens, for instance, if$X$is afinite setor more generally acompact 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 the open convex hull of a set$X$is$d$-dimensional, then every point of the hull belongs to an open convex hull of at most$2d$points of$X$.The sets of vertices of a square, regular octahedron, or higher-dimensionalcross-polytopeprovide examples where exactly$2d$points are needed.

Preservation of topological properties

Thewitch of Agnesi.The points on or above the red curve provide an example of a closed set whose convex hull is open (the openupper half-plane).

Topologically, the convex hull of anopen setis 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

$\left\{(x,y)\mathop {\bigg |} y\geq {\frac {1}{1+x^{2}}}\right\}$

(the set of points that lie on or above thewitch of Agnesi) has the openupper half-planeas its convex hull.

The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces, is generalized by theKrein–Smulian theorem,according to which the closed convex hull of a weakly compact subset of aBanach space(a subset that is compact under theweak topology) is weakly compact.

Extreme points

Main article:Krein–Milman theorem

Anextreme pointof 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 the given set, because otherwise it cannot be formed as a convex combination of given points. According to theKrein–Milman theorem,every compact convex set in a Euclidean space (or more generally in alocally 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 the open unit ball are both convex, but neither one has any extreme points.Choquet theoryextends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces.

Geometric and algebraic properties

Closure operator

The convex-hull operator has the characteristic properties of aclosure operator:

• It isextensive,meaning that the convex hull of every set$X$is a superset of$X$.
• It isnon-decreasing,meaning that, for every two sets$X$and$Y$with$X\subseteq Y$,the convex hull of$X$is a subset of the convex hull of$Y$.
• It isidempotent,meaning that for every$X$,the convex hull of the convex hull of$X$is the same as the convex hull of$X$.

When applied to a finite set of points, this is the closure operator of anantimatroid,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.

Minkowski sum

The operations of constructing the convex hull and taking theMinkowski sumcommute 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 a step towards theShapley–Folkman theorembounding the distance of a Minkowski sum from its convex hull.

Projective duality

Theprojective dualoperation 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).

Special cases

Finite point sets

Convex hull of points in the plane

Main article:Convex polytope

The convex hull of a finite point set$S\subset \mathbb {R} ^{d}$forms aconvex polygonwhen$d=2$,or more generally aconvex polytopein$\mathbb {R} ^{d}$.Each extreme point of the hull is called avertex,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$S$and that encloses all of$S$. For sets of points ingeneral position,the convex hull is asimplicial polytope.

According to theupper bound theorem,the number of faces of the convex hull of$n$points in$d$-dimensional Euclidean space is$O(n^{\lfloor d/2\rfloor })$.In particular, in two and three dimensions the number of faces is at most linear in$n$.

Simple polygons

Main article:Convex hull of a simple polygon

Convex hull ( in blue and yellow) of a simple polygon (in blue)

The convex hull of asimple polygonencloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by apolygonal chainof the polygon and a single convex hull edge, are calledpockets.Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called itsconvex 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 theErdős–Nagy theoremstates that this expansion process eventually terminates.

Brownian motion

The curve generated byBrownian motionin the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms acontinuously differentiable curve.However, for any angle$\theta$in the range$\pi /2<\theta <\pi$,there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle$\theta$.TheHausdorff dimensionof this set of exceptional times is (with high probability)$1-\pi /2\theta$.

Space curves

Anoloid,the convex hull of two circles in 3d space

For the convex hull of aspace curveor finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves aredevelopableandruled surfaces.Examples include theoloid,the convex hull of two circles in perpendicular planes, each passing through the other's center,thesphericon,the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained fromAlexandrov's uniqueness theoremfor a surface formed by gluing together two planar convex sets of equal perimeter.

Functions

Main article:Lower convex envelope

The convex hull orlower convex envelopeof a function$f$on a real vector space is the function whoseepigraphis the lower convex hull of the epigraph of$f$. It is the unique maximalconvex functionmajorized by$f$.The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from theirpointwise minimum) and, in this form, is dual to theconvex conjugateoperation.

Computation

Main article:Convex hull algorithms

Incomputational 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, efficientrepresentationof the required convex shape. Output representations that have been considered for convex hulls of point sets include a list oflinear inequalitiesdescribing thefacetsof the hull, anundirected graphof facets and their adjacencies, or the fullface latticeof 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$n$,the number of input points, and$h$,the number of points on the convex hull, which may be significantly smaller than$n$.For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis.Graham scancan compute the convex hull of$n$points in the plane in time$O(n\log n)$.For points in two and three dimensions, more complicatedoutput-sensitive algorithmsare known that compute the convex hull in time$O(n\log h)$.These includeChan's algorithmand theKirkpatrick–Seidel algorithm.For dimensions$d>3$,the time for computing the convex hull is$O(n^{\lfloor d/2\rfloor })$,matching the worst-case output complexity of the problem.The convex hull of a simple polygon in the plane can be constructed inlinear time.

Dynamic convex hulldata structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points,andkinetic convex hullstructures 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 therotating calipersmethod for computing thewidthanddiameterof a point set.

Related structures

Orthogonal convex hull

Relative convex hull

Several other shapes can be defined from 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:

• Theaffine hullis the smallest affine subspace of a Euclidean space containing a given set, or the union of all affine combinations of points in the set.
• Thelinear hullis the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set.
• Theconical hullor positive hull of a subset of a vector space is the set of all positive combinations of points in the subset.
• Thevisual hullof a three-dimensional object, with respect to a set of viewpoints, consists of the points$p$such that every ray from a viewpoint through$p$intersects the object. Equivalently it is the intersection of the (non-convex) cones generated by the outline of the object with respect to each viewpoint. It is used in3D reconstructionas the largest shape that could have the same outlines from the given viewpoints.
• The circular hull or Alpha -hull of a subset of the plane is the intersection of all disks with a given radius$1/\ Alpha$that contain the subset.
• Therelative convex hullof a subset of a two-dimensionalsimple polygonis the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains thegeodesicbetween any two of its points.
• Theorthogonal convex hullor rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally convex if it contains all axis-parallel segments between pairs of its points.
• The orthogonal convex hull is a special case of a much more general construction, thehyperconvex hull,which can be thought of as the smallestinjective metric spacecontaining the points of a givenmetric space.
• Theholomorphically convex hullis a generalization of similar concepts tocomplex analytic manifolds,obtained as an intersection of sublevel sets ofholomorphic functionscontaining a given set.

TheDelaunay triangulationof a point set and itsdual,theVoronoi diagram,are mathematically related to convex hulls: the Delaunay triangulation of a point set in$\mathbb {R} ^{n}$can be viewed as the projection of a convex hull in$\mathbb {R} ^{n+1}.$ TheAlpha shapesof 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 is the union of some of the features of the Delaunay triangulation, selected by comparing theircircumradiusto the parameter Alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. Theconvex layersof 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 the points that are not vertices of the convex hull.

Theconvex skullof a polygon is the largest convex polygon contained inside it. It can be found inpolynomial time,but the exponent of the algorithm is high.

Applications

Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to studypolynomials,matrixeigenvalues,andunitary elements,and several theorems indiscrete geometryinvolve convex hulls. They are used inrobust statisticsas the outermost contour ofTukey depth,are part of thebagplotvisualization of two-dimensional data, and define risk sets ofrandomized decision rules.Convex hulls ofindicator vectorsof solutions to combinatorial problems are central tocombinatorial optimizationandpolyhedral combinatorics.In economics, convex hulls can be used to apply methods ofconvexity in economicsto non-convex markets. In geometric modeling, the convex hull propertyBézier curveshelps 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 thehome range.

Mathematics

Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane byTverberg's theorem

Newton polygonsof univariatepolynomialsandNewton polytopesof multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze theasymptoticbehavior of the polynomial and the valuations of its roots.Convex hulls and polynomials also come together in theGauss–Lucas theorem,according to which therootsof the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.

Inspectral analysis,thenumerical rangeof anormal matrixis the convex hull of itseigenvalues. TheRusso–Dye theoremdescribes the convex hulls ofunitary elementsin aC*-algebra. Indiscrete geometry,bothRadon's theoremandTverberg's theoremconcern 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 the intersection of all convex supersets, apply tohyperbolic spacesas well as to Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets ofideal 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 toruled surfacesin Euclidean space, and their metric properties play an important role in thegeometrization conjectureinlow-dimensional topology.Hyperbolic convex hulls have also been used as part of the calculation ofcanonicaltriangulationsofhyperbolic manifolds,and applied to determine the equivalence ofknots.

See also the section onBrownian motionfor the application of convex hulls to this subject, and the section onspace curvesfor their application to the theory ofdevelopable surfaces.

Statistics

Abagplot.The outer shaded region is the convex hull, and the inner shaded region is the 50% Tukey depth contour.

Inrobust statistics,the convex hull provides one of the key components of abagplot,a method for visualizing the spread of two-dimensional sample points. The contours ofTukey depthform 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 statisticaldecision theory,the risk set of arandomized decision ruleis the convex hull of the risk points of its underlying deterministic decision rules.

Combinatorial optimization

Incombinatorial optimizationandpolyhedral combinatorics,central objects of study are the convex hulls ofindicator vectorsof solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based onlinear programmingcan be used to find optimal solutions.Inmulti-objective optimization,a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize anyquasiconvex combinationof weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.

Economics

Main article:Convexity in economics

In theArrow–Debreu modelofgeneral economic equilibrium,agents are assumed to have convexbudget setsandconvex preferences.These assumptions ofconvexity in economicscan be used to prove the existence of an equilibrium. When actual economic data isnon-convex,it can be made convex by taking convex hulls. TheShapley–Folkman theoremcan be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market.

Geometric modeling

Ingeometric modeling,one of the key properties of aBézier curveis 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 the geometry of boat and ship design,chain girthis a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of thehullof the vessel. It differs from theskin girth,the perimeter of the cross-section itself, except for boats and ships that have a convex hull.

Ethology

The convex hull is commonly known as the minimum convex polygon inethology,the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal'shome rangebased on points where the animal has been observed.Outlierscan 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 thelocal convex hullmethod by combining convex hulls ofneighborhoodsof points.

Quantum physics

Inquantum physics,thestate spaceof any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points arepositive-semidefinite operatorsknown as pure states and whose interior points are called mixed states.TheSchrödinger–HJW theoremproves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.

Thermodynamics

Convex hull ofmagnesium–carboncompounds.Mg₂C₃is expected to be unstable as it lies above the lower hull.

A convex hull inthermodynamicswas identified byJosiah Willard Gibbs(1873),although the paper was published before the convex hull was so named. In a set of energies of severalstoichiometriesof 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 the hull, its distance to the new hull represents the degree of stability of the phase.

History

The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter fromIsaac NewtontoHenry Oldenburgin 1676.The term "convex hull" itself appears as early as the work ofGarrett Birkhoff(1935), and the corresponding term inGermanappears earlier, for instance inHans Rademacher's review ofKőnig(1922). Other terms, such as "convex envelope", were also used in this time frame.By 1938, according toLloyd Dines,the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface.

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External links

• "Convex hull",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Weisstein, Eric W.,"Convex Hull",MathWorld
• "Convex Hull"byEric W. Weisstein,Wolfram Demonstrations Project,2007.