  Let $K_1, K_2$ be two compact convex sets in $\mathit{C}$. Their Minkowski
product is the set $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$. We show that the set
$K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk. Examples
for $K_1$ and $K_2$ are given so that $K_1$ and $K_2$ are triangles (including
interior) and $K_1K_2$ is not star-shaped. This gives a negative answer to a
conjecture by Puchala et. al concerning the product numerical range in the
study of quantum information science. Additional results and open problems are
presented.
