# Minkowski sum, zonotopes, convex hull

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 $$b=a+c$$.

• What are zonotopes? Are they simply simplices of dimension $\, \le 4$? Commented Oct 22, 2019 at 6:30
• @WlodAA, oh, thank you for your asking, I should have explained this. A zonotope is the Minkowski sum of some (finite) segments. A typical example is parallelograms(the Minkowski sum of two segments). (I have edited now:) Commented Oct 22, 2019 at 14:12
• Is there any constraint on $\displaystyle a$ ? if no, then the answer should be yes. based on the definition, pick \begin{gather*} b\ =\ \frac{1}{2} p_{1} +\frac{1}{2} q_{1} ,\ c\ =\ \frac{1}{2} p_{2} +\frac{1}{2} q_{2} \ \\ p_{1} ,p_{2} \ \in \ P\\ q_{1} ,q_{2} \ \in Q \end{gather*} Since $\displaystyle b,c$ and convex combination of elements from $\displaystyle P,\ Q$, then are in $\displaystyle A$, hence in $B$, Pick $\displaystyle a\ =\ \frac{1}{2}( p_{1} \ -\ p_{2}) +\ \frac{1}{2}( q_{1} -q_{2})$, Then $\displaystyle b\ =\ a+c$ Commented Jul 24, 2020 at 0:56