This paper studies local properties of Voronoi
diagrams of sets of disjoint compact convex sites in
three dimensions.

It is established that bisectors are *C*^{1} surfaces
and trisectors are *C*^{1} curves, and that as a point
moves along a trisector its clearance sphere develops
monotonically.
This monotonicity property is
useful in establishing the existence of Voronoi vertices
bounding edges in certain situations.

The paper then considers the diagram for a set of disjoint
spheres. Considerations about general
position are covered in detail.
By letting the spheres grow from point sites
till they reach their true radius, it is shown that the
Voronoi cell for the smallest site has complexity *O(n)*.
assuming that the sites are of at most *k* distinct
radii. It follows that the Voronoi diagram is
*O(n*^{2}*)*.

Although this is weaker than Aurenhammer's result
establishing *O(n*^{2}*)*
complexity with no restriction on radius, the
techniques may be of value for studying more general
Voronoi diagrams.

Finally, the paper shows that without the bound on the number of different
radii, the cell owned by a point site can have
complexity at least *n*^{2}.