Generalizations of Directly Similar Theorem? There is an attractive theorem that says that if two plane figures
are directly similar, then so is any convex combination of them.
Below, $P_1$ and $P_2$ are directly similar polygons: they have
the same angles in the same sequence;
They are (positively) scaled, rotated, translated versions of one another.
The convex combination
illustrated $P_{12} = \frac{1}{3} P_1 + \frac{2}{3} P_2$ is
also directly similar.
   
(Let me henceforth abbreviate "direcly similar" with "similar.")
My question is:

Q1.
  Is there a natural extension to $\mathbb{R}^3$ and to higher
  dimensions?

Any extension cannot be an exact extension—some aspect has to give.
For example, here are two cubes $P_1$ and $P_2$, the latter
a $\frac{3}{4}$-scale rotated version of $P_1$.
Shown is $P_{12} = \frac{1}{2} P_1 + \frac{1}{2} P_2$,
and it is clearly not a cube. 
I would define similarity in $\mathbb{R}^3$ to require all faces to
be similar with the same scale factor as well preserving all
dihedral angles—in other words, $B$ is similar to $A$ if
$B$ is a rotated, scaled, translated version of $A$.
   
But perhaps this is true?

Q2.
  Is the convex combination of two cubes (scaled, rotated,
  translated) always a parallelopiped?
  A parallelotope in $\mathbb{R}^d$?

Another approach would be to restrict the transformations:

Q3. Is there some condition on the transformations
  applied to the shapes that permits the similarity conclusion?
  Or: What is the widest class of transformations that leads
  to the similarity conclusion?

Certainly if $P_2$ is just a translated, scaled copy of $P_1$,
then any convex combination is similar, in any dimension.
Rotations are the culprit. But perhaps some rotations still
lead to similarity.
It seems likely this has all been well-explored.
If so, thanks for pointers!
 A: First of all, we may forget about translations, since they just shift the convex combinations. Thus we may regard the direct similarity as a linear transform given by an orthogonal matrix $A$ multiplied by some constant $\alpha$. Then the transform mapping the otiginal polytope to the convex combination is given by 
$$
  B=\lambda I+(1-\lambda)\alpha A.
$$ This is also a linear transform, so the image of a parallelotope is also a parallelotope (if the transform is non-degenerate; notice that it may be degenerate only if $\lambda=\pm(1-\lambda)\alpha$ since the complex eigenvalues of $A$ have unital absolute values; in this case $A$ should have eigenvalue $\pm 1$, and this may happen even in the planar case). This answers Q2.
As for Q1 (and Q3), we need to check whether $B$ is an orthogonal matrix multiplied by a real number $c$ (provided that the polytope is solid). This is the case iff all eigenvalues of $B$ have the same absolute value, which reduces to the following two options:
1) All eigenvalues of $A$ coincide (thus $A$ is scalar, and the transform is a scaling); or
2) All the eigenvalues of $A$ are $e^{\pm i\theta}$ for some fixed $\theta$ (surely these two eigenvalues coinciding multiplicities). Thus $A$ acts as a composition of rotations in $d/2$ mutually orthogonal planes. This case cannot happen in odd-dimensional space (which answers Q3), but it sometimes happens in all even-dimensional spaces. 
The phenomenon of the planar case is that for the plane there are no more options.
