| # 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);
| Example: Convex hull of the unit basis vectors in . This is the set By definition, the vector is the vector in with components . Hence, the set above is the set of vectors in with non-negative components which sum to one: where the component-wise inequality notation expresses the fact that the vector has non-negative components, and stands for the vector of ones. This set is called the probability simplex.
## 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 .