In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the convex hull or convex envelope for a set of points X in a real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V is the minimal convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

 containing X.

The convex hull also has a linear-algebraic characterization: the convex hull of X is the set of all convex combination

Convex combination

In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....

s of points in X.

In computational geometry

Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

, a basic problem is finding the convex hull for a given finite set of points in the plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

.
To show that the convex hull of a set X in a real vector space V exists, notice that X is contained in at least one convex set (the whole space V, for example), and any intersection of convex sets containing X is also a convex set containing X.