How large can the intersection of a parallelogram (or a square, if you prefer) with a triangle be, if each of them is of unit area? It is easy to see that the intersection can be of area 3/4 – is this the maximum?

The analogous question can be asked in every dimension greater than 2, replacing parallelogram with parallelepiped (or cube, if you prefer) and triangle with a simplex, each of unit volume.