Minkowski difference of two convex sets is convex?

Question

Hi my question is quite straightforward, if we have two disjoint compact convex sets A and B, is their minkowski difference A-B then convex again?

Thanks!

Answer 1

Each point in $A-B$ is of the form $a-b$, where $a\in A$ and $b\in B$.

Letting $a-b$ and $a^{\prime}-b^{\prime}$ be two points in $A-B$, we note that for any $\theta\in[0,1]$, $$ \theta\left(a-b\right)+\left(1-\theta\right)\left(a^{\prime}-b^{\prime}\right)=\left[\theta a+\left(1-\theta\right)a^{\prime}\right]-\left[\theta b+\left(1-\theta\right)b^{\prime}\right]. $$ What does this imply?