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? (It is not; see Joe's answer below. I now believe $2(\sqrt2−1)≈0.828427$ given by Joe is the correct answer.)

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 or an affine square pyramid, each of unit volume.