number of faces is at most linear in . Simple polygons The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its convex differences tree. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates. Brownian motion The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a continuously differentiable curve. However, for any angle in the range , there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle . The Hausdorff dimension of this set of exceptional times is (with high probability) . Space curves For the convex hull of a space curve or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are developable and ruled surfaces. Examples include the oloid, the convex hull of two circles in perpendicular planes, each passing through the other's center, the sphericon, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter. Functions The convex hull or lower convex envelope of a function on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of . It is the unique maximal convex function majorized by . The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation. Computation In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient