Convex Bounding Hull for N-Dimensional Point Cloud

I have an n-dimensional point cloud and I want to fit a convex bounding hull to it. The points should be the vertices of the shape. Most importantly, I want to be able to identify the points that make up the bounding hull (and, just as importantly, those that don't -- i.e. those that are inside). I don't actually care about the shape itself -- just know which points make it up and which points don't.

Computational efficiency is not as much of a problem in this one -- this isn't for real-time work.

Thoughts on the appropriate algorithm to use?

Thanks
Corey