I have a convex shape defined by a set of vertices (the so-called vertex representation of a convex polyhedron). I also have a large set of points and I would like to test which are contained in the convex shape.  Currently I just use an open source linear programming solver for each point independently with a constant objective function.  

However this is quite slow even in 100 dimensions.  Is there a way to use the fact that all the query points are known in advance to speed the process up?