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The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and $S'$ is the same square rotated $45^\circ$—it is the mean of two squares:

Of special interest to me are (a) convex polyhedra in $\mathbb{R}^3$, (b) combinations of just two polyhedra, and (c) where both are regular or semi-regular. For example, the truncated cuboctahedron (a.k.a. the great rhombicuboctahedron) "is the Minkowski sum of a cube and a truncated octahedron" (quoting from a paper ["Zonohedra and Zonotopes"] by David Eppstein). (It is also the sum of three cubes!)

My question is, essentially:

Which regular and semi-regular polyhedra can be obtained as convex combinations of pairs of other regular and semi-regular polyhedra, and which cannot be so represented—are 'prime' or unique in this respect?

I've searched around for a definitive tabulation of this information without success. It must be all known?

Particular subquestions and generalizations include:

1. Can any of the Platonic solids be realized as sums of two other Platonic solids?
2. Which of the semi-regular (Archimedean solids) can be realized as sums of two Platonic solids?
3. I hesitate to delve into the [Johnson solids][14] ...
4. Can any of the 4D regular polytopes be realized as sums of pairs of other 4D regular polytopes?
5. Can any of the three regular polytopes in dimension $d>4$ (the simplex, the hypercube, the orthoplex) be realized as sums of either of the other two?
6. Is there some general underlying theorem lurking here?

I can make (informed?) guesses and proof sketches for several of these questions, but would prefer to defer to the experts. Any observations on any aspect of these (sub)questions, references, comments, would be appreciated. Thanks!

The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and $S'$ is the same square rotated $45^\circ$—it is the mean of two squares:

Of special interest to me are (a) convex polyhedra in $\mathbb{R}^3$, (b) combinations of just two polyhedra, and (c) where both are regular or semi-regular. For example, the truncated cuboctahedron (a.k.a. the great rhombicuboctahedron) "is the Minkowski sum of a cube and a truncated octahedron" (quoting from a paper ["Zonohedra and Zonotopes"] by David Eppstein). (It is also the sum of three cubes!)

My question is, essentially:

Which regular and semi-regular polyhedra can be obtained as convex combinations of pairs of other regular and semi-regular polyhedra, and which cannot be so represented—are 'prime' or unique in this respect?

I've searched around for a definitive tabulation of this information without success. It must be all known?

Particular subquestions and generalizations include:

1. Can any of the Platonic solids be realized as sums of two other Platonic solids?
2. Which of the semi-regular (Archimedean solids) can be realized as sums of two Platonic solids?
3. I hesitate to delve into the [Johnson solids][14] ...
4. Can any of the 4D regular polytopes be realized as sums of pairs of other 4D regular polytopes?
5. Can any of the three regular polytopes in dimension $d>4$ (the simplex, the hypercube, the orthoplex) be realized as sums of either of the other two?
6. Is there some general underlying theorem lurking here?

I can make (informed?) guesses and proof sketches for several of these questions, but would prefer to defer to the experts. Any observations on any aspect of these (sub)questions, references, comments, would be appreciated. Thanks!

The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and $S'$ is the same square rotated $45^\circ$—it is the mean of two squares:

Of special interest to me are (a) convex polyhedra in $\mathbb{R}^3$, (b) combinations of just two polyhedra, and (c) where both are regular or semi-regular. For example, the truncated cuboctahedron (a.k.a. the great rhombicuboctahedron) "is the Minkowski sum of a cube and a truncated octahedron" (quoting from a paper ["Zonohedra and Zonotopes"] by David Eppstein)  . (It is also the sum of three cubes!)

My question is, essentially:

Which regular and semi-regular polyhedra can be obtained as convex combinations of pairs of other regular and semi-regular polyhedra, and which cannot be so represented—are 'prime' or unique in this respect?

I've searched around for a definitive tabulation of this information without success. It must be all known?

Particular subquestions and generalizations include:

1. Can any of the Platonic solids be realized as sums of two other Platonic solids?
2. Which of the semi-regular (Archimedean solids) can be realized as sums of two Platonic solids?
3. I hesitate to delve into the [Johnson solids][14] ...
4. Can any of the 4D regular polytopes be realized as sums of pairs of other 4D regular polytopes?
5. Can any of the three regular polytopes in dimension $d>4$ (the simplex, the hypercube, the orthoplex) be realized as sums of either of the other two?
6. Is there some general underlying theorem lurking here?

I can make (informed?) guesses and proof sketches for several of these questions, but would prefer to defer to the experts. Any observations on any aspect of these (sub)questions, references, comments, would be appreciated. Thanks!

The convex combination of convex polytopes is a convex polytope. An example in $\mathbb{R}^2$ is that a regular octagon can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$, where $S$ is a square and $S'$ is the same square rotated $45^\circ$—it is the mean of two squares:

Of special interest to me are (a) convex polyhedra in $\mathbb{R}^3$, (b) combinations of just two polyhedra, and (c) where both are regular or semi-regular. For example, the truncated cuboctahedron (a.k.a. the great rhombicuboctahedron) "is the Minkowski sum of a cube and a truncated octahedron" (quoting from a paper ["Zonohedra and Zonotopes"] by David Eppstein). (It is also the sum of three cubes!)

My question is, essentially:

Which regular and semi-regular polyhedra can be obtained as convex combinations of pairs of other regular and semi-regular polyhedra, and which cannot be so represented—are 'prime' or unique in this respect?

I've searched around for a definitive tabulation of this information without success. It must be all known?

Particular subquestions and generalizations include:

1. Can any of the Platonic solids be realized as sums of two other Platonic solids?
2. Which of the semi-regular (Archimedean solids) can be realized as sums of two Platonic solids?
3. I hesitate to delve into the [Johnson solids][14] ...
4. Can any of the 4D regular polytopes be realized as sums of pairs of other 4D regular polytopes?
5. Can any of the three regular polytopes in dimension $d>4$ (the simplex, the hypercube, the orthoplex) be realized as sums of either of the other two?
6. Is there some general underlying theorem lurking here?

I can make (informed?) guesses and proof sketches for several of these questions, but would prefer to defer to the experts. Any observations on any aspect of these (sub)questions, references, comments, would be appreciated. Thanks!