Editing Convex hull (section)

==Special cases==
===Finite point sets===
[[File:Convex hull.png|thumb|upright|Convex hull of points in the plane]]
{{main|Convex polytope}}
The convex hull of a finite point set <math>S \subset \R^d</math> forms a [[convex polygon]] when <math>d=2</math>, or more generally a [[convex polytope]] in <math>\R^d</math>. Each extreme point of the hull is called a [[Vertex (geometry)|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 <math>S</math> and that encloses all of <math>S</math>.{{sfnp|de Berg|van Kreveld|Overmars|Schwarzkopf|2008|page=3}}
For sets of points in [[general position]], the convex hull is a [[simplicial polytope]].{{sfnp|Grünbaum|2003|page=57}}

According to the [[upper bound theorem]], the number of faces of the convex hull of <math>n</math> points in <math>d</math>-dimensional Euclidean space is <math>O(n^{\lfloor d/2\rfloor})</math>.{{sfnp|de Berg|van Kreveld|Overmars|Schwarzkopf|2008|page=256}} In particular, in two and three dimensions the number of faces is at most linear in <math>n</math>.{{sfnp|de Berg|van Kreveld|Overmars|Schwarzkopf|2008|page=245}}

===Simple polygons===
{{main|Convex hull of a simple polygon}}
[[File:Convex hull of a simple polygon.svg|thumb|upright|Convex hull (in blue and yellow) of a simple polygon (in blue)]]
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 a hierarchical description of a given polygon called its ''convex differences tree''.{{sfnp|Rappoport|1992}} 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.{{sfnp|Demaine|Gassend|O'Rourke|Toussaint|2008}}

===Brownian motion===
The curve generated by [[Brownian motion]] in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a [[Differentiable curve|continuously differentiable curve]]. However, for any angle <math>\theta</math> in the range <math>\pi/2<\theta<\pi</math>, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle <math>\theta</math>. The [[Hausdorff dimension]] of this set of exceptional times is (with high probability) <math>1-\pi/2\theta</math>.{{sfnp|Cranston|Hsu|March|1989}}

===Space curves===
[[File:Oloid structure.svg|thumb|An [[oloid]], the convex hull of two circles in 3d space]]
For the convex hull of a [[space curve]] or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are [[Developable surface|developable]] and [[ruled surface]]s.{{sfnp|Sedykh|1981}} Examples include the [[oloid]], the convex hull of two circles in perpendicular planes, each passing through the other's center,{{sfnp|Dirnböck|Stachel|1997}} 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.{{sfnp|Seaton|2017}}

===Functions===
{{main|Lower convex envelope}}
The convex hull or [[lower convex envelope]] of a function <math>f</math> on a real vector space is the function whose [[Epigraph (mathematics)|epigraph]] is the lower convex hull of the epigraph of <math>f</math>.
It is the unique maximal [[convex function]] majorized by <math>f</math>.{{sfnp|Rockafellar|1970|page=36}} 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 their [[pointwise minimum]]) and, in this form, is dual to the [[convex conjugate]] operation.{{sfnp|Rockafellar|1970|page=149}}