Suppose that there exists a pentahedron in a sphere whose radius is $1$ and that each vertex of the pentahedron can exist on the surface of the sphere.
Question : What is the max of the volume of such pentahedron (if it exists)?
We can separate pentahedra into the following three types :
Type $A$ : The faces are one quadrilateral and four triangles.
Type $B$ : The faces are three quadrilaterals and two triangles which are parallel.
Type $C$ : The faces are three quadrilaterals and two triangles which are not parallel.
The followings are what I've got. Let $V(A)$, for example, be the function of the volume of type $A$.
$V(A)\le \frac{64}{81}$. The equality is attained for a square pyramid (both the length of the side of the square and the height are $\frac 43$).
$V(B)\le 1$. The equality is attained for a regular triangular prism (the length of the side of the equilateral triangles is $\sqrt 2$ and the length between the triangles is $\frac{2}{\sqrt 3}$).
However, I don't have any good idea about how to treat type $C$. Can anyone help?
P.S. 1 : I've just got another result. If a quadrilateral $ABCD$ is a rectangle and a side $EF$ is parallel to a side of the rectangle (say type $Z$), then $V(Z)\le 1$ and the equality is attained for the regular triangular prism above. Note that type $Z$ includes some of type $A$ (when $E=F$ as @Lucian comments), all of type $B$, some of type $C$.
