Editing Convex hull (section)

== Applications ==
[[File:CIE1931xy_gamut_comparison.svg|thumb|The convex hull of the primary colors in each [[color space]] on a [[CIE 1931]] xy [[chromaticity diagram]] defines the space's [[gamut]] of possible colors]]
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study [[polynomial]]s, matrix [[eigenvalue]]s, and [[unitary element]]s, and several theorems in [[discrete geometry]] involve convex hulls. They are used in [[robust statistics]] as the outermost contour of [[Tukey depth]], are part of the [[bagplot]] visualization of two-dimensional data, and define risk sets of [[randomised decision rule|randomized decision rule]]s. Convex hulls of [[indicator vector]]s 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, the convex hull property [[Bézier curve]]s 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]].

===Mathematics===
[[Newton polygon]]s of univariate [[polynomial]]s and [[Newton polytope]]s of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the [[asymptotic analysis|asymptotic]] behavior of the polynomial and the valuations of its roots.<ref>{{harvtxt|Artin|1967}}; {{harvtxt|Gel'fand|Kapranov|Zelevinsky|1994}}</ref> Convex hulls and polynomials also come together in the [[Gauss–Lucas theorem]], according to which the [[Zero of a function|roots]] of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.{{sfnp|Prasolov|2004}}

[[File:Tverberg heptagon.svg|thumb|upright|Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane by [[Tverberg's theorem]]]]
In [[Spectral theory|spectral analysis]], the [[numerical range]] of a [[normal matrix]] is the convex hull of its [[eigenvalue]]s.{{sfnp|Johnson|1976}}
The [[Russo–Dye theorem]] describes the convex hulls of [[unitary element]]s in a [[C*-algebra]].{{sfnp|Gardner|1984}}
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.{{sfnp|Reay|1979}}

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 to [[hyperbolic space]]s as well as to Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of [[ideal point]]s, 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 surface]]s in Euclidean space, and their metric properties play an important role in the [[geometrization conjecture]] in [[low-dimensional topology]].{{sfnp|Epstein|Marden|1987}} Hyperbolic convex hulls have also been used as part of the calculation of [[Canonical form|canonical]] [[Triangulation (geometry)|triangulations]] of [[hyperbolic manifold]]s, and applied to determine the equivalence of [[Knot (mathematics)|knots]].{{sfnp|Weeks|1993}}

See also the section on [[#Brownian motion|Brownian motion]] for the application of convex hulls to this subject, and the section on [[#Space curves|space curves]] for their application to the theory of [[developable surface]]s.

===Statistics===
[[File:Bagplot.png|thumb|A [[bagplot]]. The outer shaded region is the convex hull, and the inner shaded region is the 50% Tukey depth contour.]]
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.{{sfnp|Rousseeuw|Ruts|Tukey|1999}}

In statistical [[decision theory]], the risk set of a [[randomised decision rule|randomized decision rule]] is the convex hull of the risk points of its underlying deterministic decision rules.{{sfnp|Harris|1971}}

===Combinatorial optimization===
In [[combinatorial optimization]] and [[polyhedral combinatorics]], central objects of study are the convex hulls of [[indicator vector]]s 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.<ref>{{harvtxt|Pulleyblank|1983}}; see especially remarks following Theorem 2.9.</ref> In [[multi-objective optimization]], a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any [[quasiconvex function|quasiconvex combination]] of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.{{sfnp|Katoh|1992}}

===Economics===
{{main|Convexity in economics}}
In the [[Arrow–Debreu model]] of [[general equilibrium|general economic equilibrium]], agents are assumed to have convex [[budget set]]s and [[convex preferences]]. These assumptions of [[convexity in economics]] can be used to prove the existence of an equilibrium.
When actual economic data is [[Non-convexity (economics)|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.<ref>{{harvtxt|Nicola|2000}}. See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.</ref>

===Geometric modeling===
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.{{sfnp|Chen|Wang|2003}}

In 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 (watercraft)|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.{{sfnp|Mason|1908}}

===Ethology===
The convex hull is commonly known as the minimum convex polygon in [[ethology]], the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's [[home range]] based on points where the animal has been observed.<ref>{{harvtxt|Kernohan|Gitzen|Millspaugh|2001}}, p. 137–140; {{harvtxt|Nilsen|Pedersen|Linnell|2008}}</ref> [[Outlier]]s 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,{{sfnp|Worton|1995}} or in the [[local convex hull]] method by combining convex hulls of [[k-nearest neighbors algorithm|neighborhoods]] of points.{{sfnp|Getz|Wilmers|2004}}

===Quantum physics===
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 matrix|positive-semidefinite operators]] known as pure states and whose interior points are called mixed states.{{sfnp|Rieffel|Polak|2011}} The [[Schrödinger–HJW theorem]] proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.{{sfnp|Kirkpatrick|2006}}

===Thermodynamics===
[[File:Mg–C convex hull.png|thumb|Convex hull of [[magnesium]]–[[carbon]] compounds.{{sfnp|Kim|Kim|Koo|Lee|2019}} Mg<sub>2</sub>C<sub>3</sub> is expected to be unstable as it lies above the lower hull.]]
A convex hull in [[thermodynamics]] was identified by [[Josiah Willard Gibbs]] (1873),{{sfnp|Gibbs|1873}} although the paper was published before the convex hull was so named.
In a set of energies of several [[Stoichiometry|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 the hull, its distance to the new hull represents the degree of stability of the phase.<ref>{{harvtxt|Hautier|2014}}; {{harvtxt|Fultz|2020}}</ref>