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For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, parallelograms).

In the 4-dimension Euclid space, $A=\{\operatorname{conv}(P\cup Q)\mid P, Q \text{ are zonotopes}\}$, (all points and segments are zonotopes as well, thus all points, segment, trianges and tetrahedrons are in $A$).

Let $B=\{\text{any finite Minkowski sum of elements in $A$}\}$.

The question is that, for a 4-dimension simplex $a$, are there $b,c$ in $B$ such that $b = a+c$ (here $+$ is the Minkowski sum). By affine transformation, we can just consider the case where $a$ is $(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)$.

Here is my attempt: I try to consider the 3-dimension version: $A=\{\operatorname{conv}(P\cup Q)\mid P \text{ is a zonotope}, Q\text{ is a single point}\}$ and the define $B$ similarly. The question becomes: for a 3-dimension simplex $a$, are there $b,c$ in $B$ such that $b=a+c$.

For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, parallelograms).

In the 4-dimension Euclid space, $A=\{\operatorname{conv}(P\cup Q)\mid P, Q \text{ are zonotopes}\}$, (all points and segments are zonotopes as well, thus all points, segment, trianges and tetrahedrons are in $A$).

Let $B=\{\text{any finite Minkowski sum of elements in $A$}\}$.

The question is that, for a 4-dimension simplex $a$, are there $b,c$ in $B$ such that $b = a+c$ (here $+$ is the Minkowski sum). By affine transformation, we can just consider the case where the vertices of $a$ are $(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)$ (so that $a$ is the convex hull of these five points).

Here is my attempt: I try to consider the 3-dimension version: $A=\{\operatorname{conv}(P\cup Q)\mid P \text{ is a zonotope}, Q\text{ is a single point}\}$ and the define $B$ similarly. The question becomes: for a 3-dimension simplex $a$, are there $b,c$ in $B$ such that $b=a+c$.

In the 4-dimension Euclid space, $A=\{\operatorname{conv}(P\cup Q)\mid P, Q \text{ are zonotopes}\}$, (all points and segments are zonotopes as well, thus all points, segment, trianges and tetrahedrons are in $A$).

Let $B=\{\text{any finite Minkowski sum of elements in $A$}\}$.

The question is that, for a 4-dimension simplex $a$, are there $b,c$ in $B$ such that $b = a+c$ (here $+$ is the Minkowski sum). By affine transformation, we can just consider the case where $a$ is $(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)$.

Here is my attempt: I try to consider the 3-dimension version: $A=\{\operatorname{conv}(P\cup Q)\mid P \text{ is a zonotope}, Q\text{ is a single point}\}$ and the define $B$ similarly. The question becomes: for a 3-dimension simplex $a$, are there $b,c$ in $B$ such that $b=a+c$.

For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, parallelograms).

In the 4-dimension Euclid space, $A=\{\operatorname{conv}(P\cup Q)\mid P, Q \text{ are zonotopes}\}$, (all points and segments are zonotopes as well, thus all points, segment, trianges and tetrahedrons are in $A$).

Let $B=\{\text{any finite Minkowski sum of elements in $A$}\}$.

The question is that, for a 4-dimension simplex $a$, are there $b,c$ in $B$ such that $b = a+c$ (here $+$ is the Minkowski sum). By affine transformation, we can just consider the case where $a$ is $(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)$.

Here is my attempt: I try to consider the 3-dimension version: $A=\{\operatorname{conv}(P\cup Q)\mid P \text{ is a zonotope}, Q\text{ is a single point}\}$ and the define $B$ similarly. The question becomes: for a 3-dimension simplex $a$, are there $b,c$ in $B$ such that $b=a+c$.