Convex Hull - Convex Hull of A Finite Point Set

Convex Hull of A Finite Point Set

The convex hull of a finite point set is the set of all convex combinations of its points. In a convex combination, each point in is assigned a weight or coefficient in such a way that the coefficients are all positive and sum to one, and these weights are used to compute a weighted average of the points. For each choice of coefficients, the resulting convex combination is a point in the convex hull, and the whole convex hull can be formed by choosing coefficients in all possible ways. Expressing this as a single formula, the convex hull is the set:

The convex hull of a finite point set forms a convex polygon in the plane, or more generally a convex polytope in . Each point in that is not in the convex hull of the other points (that is, such that ) is called a vertex of . In fact, every convex polytope in is the convex hull of its vertices.

If the points of are all on a line, the convex hull is the line segment joining the outermost two points. When the set is a nonempty finite subset of the plane (that is, two-dimensional), we may imagine stretching a rubber band so that it surrounds the entire set and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of .

In two dimensions, the convex hull is sometimes partitioned into two polygonal chains, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for points in any dimension in general position, each facet of the convex hull is either oriented upwards (separating the hull from points directly above it) or downwards; the union of the upward-facing facets forms a topological disk, the upper hull, and similarly the union of the downward-facing facets forms the lower hull.