This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Convex hull

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The idea of convex hull of a set A is the "smallest" convex set that contains A.

The convex hull of two points is the segment that connects them. To capture this set one need some linear algebra.

While the span of vectors in a vector space is defined as the set of all their linear combinations

 a0v0 + a1v1 + ... + anvn, 
 a0, ... an are arbitrary real numbers,

the convex hull is defined as the set of all their convex combinations

 a0v0 + a1v1 + ... + anvn, 
 a0, ... an are arbitrary real numbers satisfying
 a0 + ... + an = 1.

For example, a geometric n-simplex is defined as the convex hull of n+1 points in general position:

 conv{v0, v1, ..., vn}.