$A\subset\mathbb{R}^d$ is a polytope if $A$ equals the convex hull of some finite set.
For any two sets $B$ and $C$, $B+C\equiv\{x+y:x\in B,y\in C\}$.
My question:
Let $A$ and $B$ be polytopes in $\mathbb{R}^d$. When can we find a polytope $C$ such that $A=B+C$?
Such $C$ always exists? I guess not, right?
The words you are looking for is decomposability/Minkowski decompositions/Minkowski summands. Of course, if $B = \lambda A$ for some $0<\lambda<1$, then $C=(1-\lambda)A$ will always do the job. But there are polytopes $A$ for which summands being multiples of $A$ is the only possibility. For instance, if $A$ is a simplex, then it's indecomposable, meaning that $A=B+C$ implies that $B$ and $C$ are multiples of $A$. If your question is about a characterization of decomposability, you'll find that here: Section 15.1 in B. Grünbaum: Convex Polytopes, Springer-Verlang New York, Inc. (2nd edition, 2003).
Provided you that you are given $A$ and $B$ and you know that $C$ exists such that $A=B+C$, then $C$ the Minkowski difference of $A$ and $B$, namely, $\bigcap_{b\in B}(A-b)$. Source: Lemma 3.1.11 in R. Schneider: Convex Bodies: The Brunn--Minkowski Theory, Cambridge University Press, Cambridge (2nd edition, 2013).