Polytope in Minkowski sum

Question

Is the following statement true?

Suppose that $P$ is a polytope contained in the Minkowski sum $A+B:=\{a+b: a\in A, b\in B\}$ of two convex compact sets $A$ and $B$. Then there exist polytopes $Q\subset A$ and $R\subset B$ such that $P = Q+R$.

Edit: Here's my attempt. Let $P$ be the convex hull of its extreme points $v_1,\ldots, v_k\in A+B$. Then by definition of the Minkowski sum, for $1\leq i\leq k$ there exist $a_i\in A$ and $b_i\in B$ such that $v_i = a_i + b_i$. Let $R$ be the convex hull of the $a_i$ and let $Q$ be the convex hull of the $b_i$. Then $R\subset A$ and $Q\subset B$.

Does it follow that $P=Q+R$?

Update: It only holds that $P\subset Q+R$ and compactness is not needed. I will post in the answer. Thank you.

Answer 1

Following the OP first note that $P=\text{conv}\{v_i\}_{i=1}^k = \text{conv}\{a_i+b_i\}_{i=1}^k$. Now if $z\in P=\text{conv}\{a_i+b_i\}_{i=1}^k,$ then $z$ can be written as a convex combination $z=\sum_i \lambda_i(a_i+b_i) = \sum_i \lambda_i a_i + \sum_i \lambda_i b_i \in \text{conv}\{a_i\}_{i=1}^k + \text{conv}\{b_i\}_{i=1}^k=Q+R$.

Answer 2

This isn't true. Take $A = Conv \{0, (1,0) \}$ and $B = Conv \{0, (0,1) \}$. Then $A+B$ is the unit square, and $P = Conv \{0, (1,1)\} \subset A+B$, but there is no $Q \subset A, R \subset B$ such that $P = Q + R$.