Convex and conic hull
Convex hull of a finite set of points
The convex hull of a set of points is defined as the set
By definition, this set is convex. Note the analogy with the notion of span of a collection of vectors. Here also, we consider combinations of vectors , but we restrict the weights to be non-negative and sum to one.
|
Example: Convex hull generated by six points in . Note that one of the points is in the interior of the convex hull, so that the same convex hull is generated with the remaining five points.
|
Matlab syntax to plot the convex hull (for  )
>> inds = convhull(x,y);
>> plot(x,y);
Convex hull of a set
More generally, for any given set in , we can define its convex hull as the set of convex combinations of any finite collection of points contained in it.
|
Example: The convex hull of the union of two ellipses.
|
Conic hull
The conic hull of a set of points is defined as
Example: The conic hull of the union of the three-dimensional simplex above and the singleton is the whole set , which is the set of real vectors that have non-negative components. The figure shows that indeed any vector can be obtained by a positive scaling of a vector in the three-dimensional simplex: , with .
|