This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of unit area.

Now I ask:

Question.Suppose a parallelogram is of area 1 (say, the unit square if you prefer) and a triangle is of area $1$ as well, then what is the minimum area of the convex hull of the union of the parallelogram and the triangle?

It is obvious that the minimum exists and is greater than $1$. The best I can do is $\sqrt{2}$ $-$ is this the minimum?

My example is: the unit square and the right triangle with legs of length $\sqrt2$ each, the triangle's right angle coinciding with the square's corner. (Added by Joe's request.)

In fact, there is a continuous cyclic family of examples, each producing the same area of the convex hull:

An analogous question can be asked in three dimensions, replacing *parallelogram* with *parallelepiped* (or *cube*, if you prefer) and *triangle* with *simplex* or *pyramid with a parallelogram base*, each of unit volume.