A while ago I wrote mainly about OBB algorithms but a little note about convex hull simplification
It's a little unclear, so I clarified :
My algorithm is very simple and by no means optimal.
I construct a standard (exact) convex hull, then make a mesh from it. I then run a normal mesh simplifier (see for example Garland Heckbert Quadric Error Metrics) to simplify the CH as if it was a mesh. This can ruin inclusion. I then fix it by taking all the face planes of the simplified mesh and pushing them out past any vert in the original mesh.
Stan's (Melax - Convex Hull Simplification With Containment By Successive Plane Removal) way is similar but better. He uses a BSP engine to create the hull. First he finds a normal convex hull. Then he considers only the planes that make up that hull. The working hull is the volume that is on the "front" side of all planes. He then considers removing planes one by one. When you remove a plane, the cost to remove it is the volume that is added to the hull, which is the volume of the space that is on the back side of that plane but is on the front side of all other planes. You create a heap to do this so that the total cost to simplify is only N log N. This requires good BSP code which I don't have, which is why I used the mesh-simplifier approach.
An alternative in the literature is the "progressive hull" technique. This is basically using PM methods but directly considering the mesh as a hull during simplification instead of fixing it after the fact as I do. Probably a better way is to use a real epsilon-hull finder from the beginning rather than finding the exact hull and then simplifying.
My code is in Galaxy4 / gApp_HullTest which is available here ; You should be able to run "Galaxy4.exe hull" ; Hit the "m" key to see various visualations ; give it a mesh argument if you have one (takes .x, .m , .smf etc.)
BTW to summarize : I don't really recommend my method. It happens to be easy to implement if you have a mesh simplifier lying around. Stan's method is also certainly not optimal but is easy to implement if you have good BSP code lying around (and is better than mine (I suspect)).
The technique I actually prefer is to just use k-dops. k-dops are the convex hull made from the touching planes in a fixed set of k directions. Maybe find the optimal OBB and use that as the axis frame for the k directions. Increase k until you are within the desired error tolerance (or k exceeds the number of faces in the exact hull).
ASIDE : I have some BSP code but I hate it; I hate all floating point geometry code. I love integer geometry code. The problem with integers in BSP's is that clipping creates rational points. Maybe I'll write some BSP routines based on rational Vec3's. The only big problem is that the precision requirement goes up with each clip. So you either need arbitrary precision rationals or you have to truncate the precision at some maximum, and then handle the errors created by that (like the truncated point could move onto the back side of a plane that you said you were in front of). (this is better than the errors in floating points, because at least the truncated point is at a definite well defined location, floating points move around depending on how you look at them, those wiggly bastards) (I'm tempted to say that they're like quantum mechanics in that they change when you measure them, except that they really aren't at all, and that's the type of pseudo-scientific-mumbo-jumbo that pseudo-intellectual fucktards love and I so despise, so no, I won't say it).