Editing Convex hull (section)

== Related structures ==
{{multiple image
|image1=Orthogonal-convex-hull.svg
|caption1=[[Orthogonal convex hull]]
|image2=Relative convex hull.svg
|caption2=[[Relative convex hull]]
|total_width=450
}}
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:
*The [[affine hull]] is the smallest affine subspace of a Euclidean space containing a given set, or the union of all affine combinations of points in the set.{{sfnp|Westermann|1976}}
*The [[linear hull]] is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set.{{sfnp|Westermann|1976}}
*The [[conical hull]] or positive hull of a subset of a vector space is the set of all positive combinations of points in the subset.{{sfnp|Westermann|1976}}
*The [[visual hull]] of a three-dimensional object, with respect to a set of viewpoints, consists of the points <math>p</math> such that every ray from a viewpoint through <math>p</math> 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 in [[3D reconstruction]] as the largest shape that could have the same outlines from the given viewpoints.{{sfnp|Laurentini|1994}}
*The circular hull or alpha-hull of a subset of the plane is the intersection of all disks with a given radius <math>1/\alpha</math> that contain the subset.{{sfnp|Edelsbrunner|Kirkpatrick|Seidel|1983}}
*The [[relative convex hull]] of a subset of a two-dimensional [[simple polygon]] is the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains the [[geodesic]] between any two of its points.{{sfnp|Toussaint|1986}}
*The [[orthogonal convex hull]] or 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.{{sfnp|Ottmann|Soisalon-Soininen|Wood|1984}}
*The orthogonal convex hull is a special case of a much more general construction, the [[hyperconvex hull]], which can be thought of as the smallest [[injective metric space]] containing the points of a given [[metric space]].{{sfnp|Herrlich|1992}}
*The [[holomorphically convex hull]] is a generalization of similar concepts to [[complex analytic manifold]]s, obtained as an intersection of sublevel sets of [[holomorphic functions]] containing a given set.{{sfnp|Rossi|1961}}

The [[Delaunay triangulation]] of a point set and its [[dual (mathematics)|dual]], the [[Voronoi diagram]], are mathematically  related to convex hulls: the Delaunay triangulation of a point set in <math>\R^n</math> can be viewed as the projection of a convex hull in <math>\R^{n+1}.</math>{{sfnp|Brown|1979}}
The [[alpha shape]]s 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 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.{{sfnp|Edelsbrunner|Kirkpatrick|Seidel|1983}}
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 the points that are not vertices of the convex hull.{{sfnp|Chazelle|1985}}

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.{{sfnp|Chang|Yap|1986}}