3d - Collision detection of convex shapes on voxel terrain

I have some standard convex shapes (cubes, capsules) on a voxel terrain. It is very easy to detect single vertex collisions. However, it becomes computationally expensive when many vertices are involved.

To clarify, currently my algorithm represents a cube as multiple vertices covering every face of the cube, not just the corners. This is because the cubes can be much bigger than the voxels, so multiple sample points (vertices) are required (the distance between sample points must be at least the width of a voxel). This very rapidly becomes intractable.

It would be great if there were some standard algorithm(s) for collision detection between convex shapes and arbitrary voxel based terrain (like there is with OBB's and seperating axis theorem etc).

Any help much appreciated.