think about what information needs to be represented from that triangle, if you want to test if a point is inside your convex hull, then you need to test that it's inside all the planes of each triangle, if you want to test ray intersection with your triangle, first you have to get the point's intersection point with the plane that the triangle resides on, then you have to test that the point is inside the triangle.
Testing whether a point is inside a convex mesh takes O(log(n)) time. Why test it against all triangles?
Same thing goes for intersecting a ray with a convex mesh. Testing it against all triangles is suboptimal (can be done in O(log(n)) time).