Minkowski sum and Polygons

Question

The problem:..

Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that:

every vertex $p \in A + B$ is a Minkowski sum of vertices of $A$ and $B$

Hint: Use the external perpendicular to $u$.

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My attempt:

We could use that lemma:

for every external perpendicular u to an edge of A, there exists an external perpendicular to an edge of A + B, which will be parallel to u.

which I proved in another time.

Answer 1

Using the hint, let two adjacent edges $e,f$ of $A$ which meet at vertex $a$ have external perpendiculars $u,v$ (which are obviously not parallel; this would correspond to a discontinuity of the tangent to the perimeter in my previous suggestion, which ignores the hint you are given). Now use the lemma on these two edges and their external perpendiculars $u$ & $v$. All that remains is to argue that the corresponding edges in $A+B$ meet at a vertex. But wouldn't that have to be true if the images of $e$ & $f$ are connected and the corresponding external perpendiculars change from $u$ to $v$?