**Deleting points**

Let the new convex hull be obtained by deleting *nd* tetrahedra from and added *na* tetrahedra to the old convex hull. If the interior point *o* (needed for a tetrahedronization of a convex polytope), after several deletions, lies inside the new convex hull, then the same formulas and time complexity, as by adding points, follow. If *o* lie outside the new convex hull, then, we need to choose a new interior point *o*� , and recompute the new tetrahedra associated with it. Thus, we need in total *O*(*n*) time to update the principal components.

Under certain assumptions, we can recompute the new principal components faster:


Note that in the case when we consider boundary of a convex polyhedron (Subsection 6.1 and Subsection 6.3), we do not need an interior point *o* and the same time complexity holds for both adding and deleting points.
